Thai O-NET Tests - คณิตศาสตร์ - 2021 ข้อ 4

With answers from 41 AI models.

Back to Thai O-NET Tests

Question

กำหนดให้ U = {1,2,3,...,50} และ A = {neu | n เป็นจำนวนคู่} B = {neu | n หารด้วย 3 ลงตัว} จำนวนสมาชิกของ A – (A^B) เท่ากับเท่าใด

  1. 8
  2. 9
  3. 15
  4. 17
  5. 19
D is the correct answer according to the dataset.

Using dataset from OpenThaiGPT Evaluation Dataset.

Answers by AI

Let me solve this step by step:

  1. First, let's identify set A:

    • A is the set of even numbers up to 50
    • A = {2,4,6,8,...,48,50}
    • A has 25 members

  2. Next, let's identify set B:

    • B is the set of multiples of 3 up to 50
    • B = {3,6,9,12,...,45,48}
    • B has 16 members

  3. A∩B (intersection) is:

    • Numbers that are both even AND multiples of 3
    • A∩B = {6,12,18,24,30,36,42,48}
    • A∩B has 8 members

  4. A-(A∩B) means:

    • Elements in A that are not in the intersection
    • This is even numbers that are not multiples of 3
    • Total = number of elements in A minus number of elements in A∩B
    • 25 - 8 = 17

Therefore, A-(A∩B) has 17 elements.

{"correct_answer_key":"d"}

I need to find the number of elements in A - (A∩B).

First, let's understand the sets:

  • U = {1, 2, 3, ..., 50} is the universal set
  • A = {n∈U | n is even} = {2, 4, 6, ..., 50}, which contains 25 elements
  • B = {n∈U | n is divisible by 3} = {3, 6, 9, ..., 48}, which contains 16 elements

A∩B contains elements that are both even and divisible by 3, which means they are divisible by 6. A∩B = {6, 12, 18, ..., 48}, which contains 8 elements.

A - (A∩B) contains elements that are in A but not in A∩B, which means even numbers that are not divisible by 6. |A - (A∩B)| = |A| - |A∩B| = 25 - 8 = 17

Therefore, the number of elements in A - (A∩B) is 17.

{"correct_answer_key":"d"}

Let's break down this problem. We're given:

  • $U = {1, 2, 3, \ldots, 50}$ is the universal set.
  • $A = {n \in U : n \text{ is even}}$
  • $B = {n \in U : n \text{ is divisible by 3}}$

We need to find the number of elements in $A - (A \cap B)$.

First, let's identify what $A$ is. $A$ is the set of even numbers from 1 to 50. These are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50. There are 25 even numbers from 1 to 50.

Next, let's identify what $B$ is. $B$ is the set of numbers divisible by 3 from 1 to 50. These are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48. There are 16 numbers divisible by 3 from 1 to 50.

Now, we need to find $A \cap B$, which is the set of numbers that are both even and divisible by 3. These are numbers divisible by 6 (since 6 is the LCM of 2 and 3). From 1 to 50, the numbers divisible by 6 are: 6, 12, 18, 24, 30, 36, 42, 48. There are 8 numbers divisible by 6 from 1 to 50.

Finally, we can calculate $A - (A \cap B)$, which is the set of numbers that are in $A$ but not in the intersection of $A$ and $B$. This is equivalent to $A \cap B^c$. Number of elements in $A - (A \cap B) = |A| - |A \cap B| = 25 - 8 = 17$.

So, the number of elements in $A - (A \cap B)$ is 17.

Actually, wait, I'm noticing the notation in the problem is different than I initially thought. It says:

  • $A = {n \in U : n \text{ is even}}$
  • $B = {n \in U : n \text{ is divisible by 3}}$

But the notation uses "neu". Let me re-read the problem.

I think I misinterpreted the sets. Let me reread:

  • $A = {neu | n \text{ is even}}$
  • $B = {neu | n \text{ is divisible by 3}}$

Actually, it looks like the sets are defined with a specific formula, not just conditions. Let's break it down again:

  • $A = {neu | n \text{ is even}}$. This means $A$ consists of numbers of the form "neu" where $n$ is even. I'm not entirely sure what "neu" means, but it might be a notation for some operation involving $n$, $e$, and $u$.

  • $B = {neu | n \text{ is divisible by 3}}$. Similarly, $B$ consists of numbers of the form "neu" where $n$ is divisible by 3.

Hmm, I'm not familiar with the notation "neu" in this context. Let me try to interpret it in the most straightforward way.

Let me actually calculate $A - (A \cap B)$ differently. The set $A - (A \cap B)$ consists of elements that are in $A$ but not in $B$. In other words, it's the set of numbers of the form "neu" where $n$ is even but not divisible by 3.

For a number to be even but not divisible by 3, it must be of the form $n = 2m$ where $m$ is not a multiple of 3. Equivalently, the number modulo 6 must be 2 or 4.

Let's count the numbers from 1 to 50 that are even but not divisible by 3: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50.

That gives us 17 numbers. So, the number of elements in $A - (A \cap B)$ is 17.

Actually, I think I need to understand the notation more clearly.

I just realized I may have misinterpreted the notation. Let's see if there's another way to interpret it.

It might be that "neu" is not a formula but just a representation of an element, and the condition "$n$ is even" is the only condition. If that's the case, then $A$ is indeed the set of even numbers from 1 to 50, and $B$ is the set of numbers divisible by 3 from 1 to 50, as I had initially thought.

Let's go with this interpretation for now, and revisit if needed.

As I calculated earlier, there are 25 even numbers from 1 to 50, and there are 8 numbers from 1 to 50 that are both even and divisible by 3. Therefore, the number of elements in $A - (A \cap B)$, which are the even numbers from 1 to 50 that are not divisible by 3, is $25 - 8 = 17$.

So, the number of elements in $A - (A \cap B)$ is 17.

Actually, wait. I'm thinking there's a typo here, but I could be wrong. Let's interpret the notation more carefully.

Actually let me re-read the problem one more time.

I think I was overthinking this. The notation "neu" likely just means "$n$ such that". So, the sets are defined as:

  • $A = {n \in U : n \text{ is even}}$
  • $B = {n \in U : n \text{ is divisible by 3}}$

Let's proceed with this interpretation.

Now, we need to calculate $A - (A \cap B)$. This is the set of even numbers from 1 to 50 that are not divisible by 3.

  • $A$: Even numbers from 1 to 50. These are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50. There are 50/2 = 25 of them.

  • $A \cap B$: Even numbers from 1 to 50 that are divisible by 3. These are: 6, 12, 18, 24, 30, 36, 42, 48. There are 8 of them. These are the numbers divisible by 6 (LCM of 2 and 3).

  • $A - (A \cap B)$: Even numbers from 1 to 50 that are not divisible by 3. These are: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50. There are 25 - 8 = 17 of them.

So, the number of elements in $A - (A \cap B)$ is 17.

So the correct answer is "d".

Actually, hm. I should redo this to make sure I'm interpreting the notation correctly. Let me just state the problem in a way that makes sense to me, and let me know if I'm not following you interpretation:

$U$ is the set of integers from 1 to 50 inclusive. $A$ is the subset of $U$ consisting of the even integers. $B$ is the subset of $U$ consisting of integers that are divisible by 3.

The question asks: what is the size of the set $A - (A \cap B)$?

As I calculated already, there are 25 even numbers from 1 to 50 (there are 50 numbers in total, half of which are even). There are 8 numbers from 1 to 50 that are both even and divisible by 3 (these are the multiples of 6). Therefore, the number of even numbers that are not divisible by 3 is 25 - 8 = 17.

So, the number of elements in $A - (A \cap B)$ is 17, which corresponds to option "d".

Actually, wait, I might have misunderstood the notation. I'm interpreting this as:

  • $A = {n \in U | n \text{ is even}}$
  • $B = {n \in U | n \text{ is divisible by 3}}$

But the notation has "neu".

Wait! there's a logical mistake here.

The notation is:

  • $A = {neu | n \text{ is even}}$
  • $B = {neu | n \text{ is divisible by 3}}$

Looking at it carefully, "neu" might be a specific value or formula involving $n$, $e$, and $u$. It's not just a placeholder for an element in general.

But on second thought, it's also possible that "neu" is a shorthand for "n∈U", which aligns with my previous interpretation.

Actually, I think Thai notation is being used here:

"neu" might be Thai shorthand for "n is an element of set U" (similar to notation like "n∈U" in standard set theory).

Given this interpretation, $A$ would indeed be the set of even numbers from 1 to 50, and $B$ would be the set of numbers divisible by 3 from 1 to 50.

Let's recalculate to make sure:

  • $|A|$ = 25 (even numbers from 1 to 50)
  • $|B|$ = 16 (numbers divisible by 3 from 1 to 50)
  • $|A \cap B|$ = 8 (numbers divisible by 6 from 1 to 50)

Then, $|A - (A \cap B)| = |A| - |A \cap B| = 25 - 8 = 17$.

So the number of elements in $A - (A \cap B)$ is 17, which corresponds to option "d".

Let's double-check by directly calculating $A - (A \cap B)$:

$A$ = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50}

$A \cap B$ = {6, 12, 18, 24, 30, 36, 42, 48}

$A - (A \cap B)$ = {2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50}

This gives us 17 elements.

So, the answer is option "d": 17.

Actually, I'm a bit uncertain about my interpretation of the notation. Let me reconsider.

In the context of set theory, especially when working with universal sets, "n∈U" is a common notation to indicate that n is an element of set U. In this problem, the notation "neu" might be a shorthand for this, especially if the problem is presented in a language other than English, and "neu" is an abbreviation in that language.

But if "neu" is indeed a specific formula involving n, e, and u, then my interpretation might be incorrect. I'd need more context or clarity on what "neu" represents.

Given the options provided, I'll assume my interpretation of "neu" as a shorthand for "n∈U" is correct, leading to an answer of 17 (option "d").

Let me continue with this assumption and approach the problem one more time to make sure I get the answer right.

We have:

  • $U = {1, 2, 3, \ldots, 50}$
  • $A = {n \in U | n \text{ is even}}$
  • $B = {n \in U | n \text{ is divisible by 3}}$

To find $A - (A \cap B)$, we first identify the elements of sets $A$ and $B$:

$A$ = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50}

$B$ = {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48}

Then, $A \cap B$, the intersection of sets $A$ and $B$, consists of numbers that are both even and divisible by 3: $A \cap B$ = {6, 12, 18, 24, 30, 36, 42, 48}

Now, $A - (A \cap B)$ is the set of elements in $A$ that are not in $A \cap B$: $A - (A \cap B)$ = {2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50}

Counting these elements, we get 17.

Therefore, the number of elements in $A - (A \cap B)$ is 17, which corresponds to option "d".

I'm fairly confident in this answer based on my interpretation of the problem.

So the answer is d, 17.

The problem asks for the number of elements in A - (A∩B) where:

  • U = {1,2,3,...,50}
  • A = {n∈U | n is even}
  • B = {n∈U | n is divisible by 3}

First, I'll identify the sets:

  • Set A contains all even numbers from 1 to 50: 2, 4, 6, ..., 50 (25 numbers)
  • Set A∩B contains numbers that are both even and divisible by 3, which are the multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 (8 numbers)

The set A - (A∩B) contains elements that are in A but not in A∩B, which are even numbers that are not divisible by 3. |A - (A∩B)| = |A| - |A∩B| = 25 - 8 = 17

{"correct_answer_key":"d"}

To solve this problem, let's break it down step by step.

  1. Define the Sets:

    • U is the universal set containing integers from 1 to 50.
    • A is the set of even numbers in U.
    • B is the set of numbers in U that are divisible by 3.

  2. Identify Elements of A and B:

    • A (even numbers): {2, 4, 6, ..., 50}
    • B (numbers divisible by 3): {3, 6, 9, ..., 48}

  3. Find the Intersection A ∩ B:

    • A ∩ B consists of numbers that are both even and divisible by 3, i.e., multiples of 6.
    • A ∩ B = {6, 12, 18, 24, 30, 36, 42, 48}

  4. Find A - (A ∩ B):

    • This is the set of even numbers that are not divisible by 6.
    • A - (A ∩ B) = {2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50}

  5. Count the Elements in A - (A ∩ B):

    • The elements are: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50.
    • There are 17 elements.

Thus, the number of elements in A - (A ∩ B) is 17.

{"correct_answer_key":"d"}

Explanation:

Given:

  • Universal set ( U = {1, 2, 3, \dots, 50} )
  • ( A = {n \in U \mid n \text{ is even}} ) (all even numbers in ( U ))
  • ( B = {n \in U \mid n \text{ is divisible by 3}} ) (all multiples of 3 in ( U ))

We need to find the number of elements in ( A - (A \cap B) ), which is equivalent to ( |A| - |A \cap B| ).

Step 1: Find ( |A| ) (Number of even numbers in ( U ))

  • The set ( A ) contains all even numbers from 1 to 50.
  • The count is ( \frac{50}{2} = 25 ).

Step 2: Find ( |A \cap B| ) (Numbers that are even and divisible by 3)

  • ( A \cap B ) consists of numbers divisible by both 2 and 3, i.e., divisible by 6.
  • The count is ( \left\lfloor \frac{50}{6} \right\rfloor = 8 ) (since ( 6 \times 8 = 48 \leq 50 )).

Step 3: Compute ( |A - (A \cap B)| )

  • This is the number of elements in ( A ) that are not in ( A \cap B ).
  • ( |A - (A \cap B)| = |A| - |A \cap B| = 25 - 8 = 17 ).

Final Answer:

The correct answer is D (17).

{"correct_answer_key":"d"}

Okay, let's try to solve this problem. So, the question is in Thai, but I can translate it. It says:

Given U = {1,2,3,...,50} and A = {n ∈ U | n is an even number} B = {n ∈ U | n is divisible by 3} Find the number of elements in A – (A ∩ B), which is the same as A minus the intersection of A and B. The options are a)8, b)9, c)15, d)17, e)19.

Alright, first, let's understand what A – (A ∩ B) means. In set theory, A minus the intersection of A and B is equivalent to the elements that are in A but not in B. Because A ∩ B is the set of elements that are in both A and B, so subtracting that from A leaves us with elements in A that are not in B. So, essentially, we need to find the number of even numbers in U (which is A) that are not divisible by 3 (since B is numbers divisible by 3).

So, the problem reduces to finding |A| - |A ∩ B|, which is the same as counting the even numbers from 1 to 50 that are not divisible by 3.

Let's break it down step by step.

First, find the number of elements in set A. A is all even numbers from 1 to 50. Since 50 is even, the count is 50/2 = 25. So |A| = 25.

Next, find the number of elements in A ∩ B. These are numbers that are even and divisible by 3. So numbers divisible by both 2 and 3, which means divisible by 6. So we need to count the numbers from 1 to 50 that are divisible by 6.

The largest number ≤50 divisible by 6 is 48 (since 6*8=48). The smallest is 6. So the numbers are 6, 12, 18, ..., 48. This is an arithmetic sequence with first term 6, common difference 6. The number of terms is given by (48 - 6)/6 + 1 = (42/6) +1 = 7 +1 = 8. So |A ∩ B| = 8.

Therefore, the number of elements in A – (A ∩ B) is |A| - |A ∩ B| = 25 - 8 = 17.

Looking at the options, d)17 is the answer. Let me double-check to make sure I didn't make a mistake.

Wait, let me verify the count of numbers divisible by 6 up to 50 again. 6*8=48, which is correct. So numbers are 6,12,18,24,30,36,42,48. That's 8 numbers. Yes, that's right. So 25 -8=17. So the answer should be d)17.

The problem asks for the number of elements in set A minus the intersection of A and B (A – (A ∩ B)). This is equivalent to counting even numbers in U (A) that are not divisible by 3 (since B is numbers divisible by 3).

  1. |A|: Number of even numbers in U (1 to 50) = 25 (since 50/2 = 25).
  2. |A ∩ B|: Numbers divisible by both 2 and 3 (i.e., divisible by 6). The largest multiple of 6 ≤ 50 is 48 (6 × 8). Thus, there are 8 such numbers (6, 12, 18, ..., 48).
  3. A – (A ∩ B): 25 (total even numbers) – 8 (even numbers divisible by 3) = 17.
{"correct_answer_key":"d"}

The question is in Thai and asks about set operations. Here's the translation and solution:

  • U = {1, 2, 3, ..., 50} (The universal set)
  • A = {n ∈ U | n is an even number} (Set A contains even numbers)
  • B = {n ∈ U | n is divisible by 3} (Set B contains numbers divisible by 3)

The question asks for the number of elements in A - (A∩B). This represents the elements in A that are not in the intersection of A and B. In other words, we want the even numbers that are not divisible by 3.

  1. Elements in A: There are 50/2 = 25 even numbers in U.
  2. Elements in A∩B: These are numbers divisible by both 2 and 3, meaning they are divisible by 6. There are 50/6 = 8 (with a remainder, but we only take the whole number part) such numbers.
  3. Elements in A - (A∩B): This is the number of elements in A minus the number of elements in the intersection: 25 - 8 = 17

Therefore, the answer is 17.

{"correct_answer_key": "d"}

The question is in Thai. Let's translate and solve it:

Translation:

Given U = {1, 2, 3, ..., 50} A = {n ∈ U | n is even} B = {n ∈ U | n is divisible by 3}

What is the number of elements in A - (A ∩ B)?

Explanation:

  • U: The universal set containing integers from 1 to 50.
  • A: The set of even numbers within U.
  • B: The set of numbers within U that are divisible by 3.
  • A ∩ B: The intersection of A and B, meaning the set of numbers within U that are both even and divisible by 3. This is equivalent to the set of numbers divisible by 6.
  • A - (A ∩ B): The set difference between A and (A ∩ B). This means the set of even numbers in U that are not divisible by 6.

Now let's calculate the number of elements in each set:

  • |A|: Number of even numbers from 1 to 50 is 50/2 = 25.
  • |A ∩ B|: Number of multiples of 6 from 1 to 50 is floor(50/6) = 8.
  • |A - (A ∩ B)|: |A| - |A ∩ B| = 25 - 8 = 17.

Therefore, the number of elements in A - (A ∩ B) is 17.

{"correct_answer_key":"d"}

Let's break down the problem:

  • U: The universal set containing integers from 1 to 50.
  • A: The set of even numbers within U.
  • B: The set of multiples of 3 within U.
  • A ∩ B: The intersection of A and B, meaning the set of numbers that are both even and multiples of 3 (i.e., multiples of 6).
  • A - (A ∩ B): The set difference, meaning the elements in A that are not in (A ∩ B). This is equivalent to the even numbers that are not multiples of 6.

Calculations:

  1. |A| (Number of even numbers): There are 50/2 = 25 even numbers from 1 to 50.
  2. |A ∩ B| (Number of multiples of 6): The multiples of 6 within U are 6, 12, 18, ..., 48. There are 48/6 = 8 multiples of 6.
  3. |A - (A ∩ B)|: This is the number of even numbers minus the number of multiples of 6. So, 25 - 8 = 17.

Therefore, the number of elements in A - (A ∩ B) is 17.

{"correct_answer_key":"d"}

คำอธิบาย: คำถามกำหนดเซ็ต U เป็นเซ็ตของจำนวนเต็มตั้งแต่ 1 ถึง 50, เซ็ต A เป็นเซ็ตของจำนวนคู่ใน U, และเซ็ต B เป็นเซ็ตของจำนวนที่หารด้วย 3 ลงตัวใน U คำถามต้องการหาจำนวนสมาชิกของเซ็ต A ลบด้วย (A อินเตอร์เซก B) ซึ่งหมายถึงจำนวนสมาชิกในเซ็ต A ที่ไม่อยู่ในเซ็ต B

ขั้นตอนการคำนวณ:

  1. หาจำนวนสมาชิกของเซ็ต A (|A|): เซ็ต A คือจำนวนคู่ใน U = {1, 2, 3, ..., 50} จำนวนคู่ใน U คือ {2, 4, 6, ..., 50} จำนวนสมาชิกของ A คือ 50/2 = 25 สมาชิก

  2. หาเซ็ต A อินเตอร์เซก B (A ∩ B): เซ็ต A ∩ B คือจำนวนที่อยู่ในทั้ง A และ B นั่นคือจำนวนที่ทั้งเป็นจำนวนคู่และหารด้วย 3 ลงตัว ซึ่งหมายถึงจำนวนที่หารด้วย ค.ร.น. ของ 2 และ 3 คือ 6 ลงตัว จำนวนที่หารด้วย 6 ลงตัวใน U คือ {6, 12, 18, ..., 48}

  3. หาจำนวนสมาชิกของเซ็ต A อินเตอร์เซก B (|A ∩ B|): จำนวนสมาชิกของ A ∩ B คือจำนวนพหุคูณของ 6 ที่ไม่เกิน 50 จำนวนเหล่านี้คือ 6, 12, 18, 24, 30, 36, 42, 48 มีทั้งหมด 8 สมาชิก

  4. หาเซ็ต A ลบด้วย (A ∩ B) หรือ A - (A ∩ B): เซ็ต A - (A ∩ B) คือเซ็ตของสมาชิกที่อยู่ใน A แต่ไม่อยู่ใน (A ∩ B) ซึ่งก็คือเซ็ตของจำนวนคู่ที่ไม่หารด้วย 3 ลงตัว

  5. หาจำนวนสมาชิกของเซ็ต A - (A ∩ B) (|A - (A ∩ B)|): จำนวนสมาชิกของ A - (A ∩ B) สามารถคำนวณได้จาก |A| - |A ∩ B| = 25 - 8 = 17 สมาชิก

ดังนั้น จำนวนสมาชิกของ A – (A ∩ B) เท่ากับ 17 ซึ่งตรงกับตัวเลือก d.

{"correct_answer_key":"d"}

Here's the breakdown of the problem: The universal set U contains integers from 1 to 50. Set A contains even numbers in U. Set B contains numbers divisible by 3 in U. We need to find the number of elements in A - (A ∩ B).

A - (A ∩ B) is the set of elements that are in A but not in A ∩ B. This is equivalent to A - B, which means elements that are in A but not in B. Elements in A are even numbers. Elements in B are multiples of 3. A ∩ B contains elements that are both even and multiples of 3, meaning they are multiples of 6.

  1. Find the number of elements in A (|A|): Even numbers from 1 to 50 are 2, 4, ..., 50. There are 50/2 = 25 elements. So, |A| = 25.
  2. Find the number of elements in A ∩ B (|A ∩ B|): Multiples of 6 from 1 to 50 are 6, 12, ..., 48. The largest multiple of 6 less than or equal to 50 is 48 (6 * 8). So, there are 8 elements. |A ∩ B| = 8.
  3. Calculate |A - (A ∩ B)| = |A| - |A ∩ B| = 25 - 8 = 17.

The number of elements in A - (A ∩ B) is 17. This corresponds to option d.

{"correct_answer_key":"d"}

The question asks for the number of elements in the set A - (A ∩ B), where U is the set of integers from 1 to 50, A is the set of even numbers in U, and B is the set of numbers divisible by 3 in U. The expression A - (A ∩ B) is equivalent to A - B, which represents the elements in A that are not in B. In this context, it means the even numbers between 1 and 50 that are not divisible by 3.

First, let's find the number of elements in A (even numbers from 1 to 50). The even numbers are 2, 4, 6, ..., 50. This is an arithmetic sequence with the first term 2, common difference 2, and last term 50. The number of terms is (50 - 2)/2 + 1 = 48/2 + 1 = 24 + 1 = 25. So, |A| = 25.

Next, let's find the number of elements in A ∩ B (numbers that are both even and divisible by 3, i.e., divisible by 6, from 1 to 50). The numbers divisible by 6 are 6, 12, 18, ..., 48. This is an arithmetic sequence with the first term 6, common difference 6, and last term 48. The number of terms is (48 - 6)/6 + 1 = 42/6 + 1 = 7 + 1 = 8. So, |A ∩ B| = 8.

Finally, the number of elements in A - (A ∩ B) is |A| - |A ∩ B| = 25 - 8 = 17.

{"correct_answer_key":"d"}

Explanation:

  1. Identify the sets:

    • U = {1, 2, 3, ..., 50} (Universal set)
    • A = {n ∈ U | n is an even number}. The number of elements in A is n(A) = 50 / 2 = 25.
    • B = {n ∈ U | n is divisible by 3}. The number of elements in B is n(B) = floor(50 / 3) = 16.

  2. Identify the intersection A ∩ B:

    • A ∩ B contains elements that are in both A and B. These are numbers in U that are both even and divisible by 3, meaning they are divisible by 6.
    • A ∩ B = {6, 12, 18, 24, 30, 36, 42, 48}.
    • The number of elements in A ∩ B is n(A ∩ B) = floor(50 / 6) = 8.

  3. Calculate the number of elements in A - (A ∩ B):

    • A - (A ∩ B) represents the elements that are in A but not in the intersection A ∩ B.
    • The number of elements in this set is n(A) - n(A ∩ B).
    • n(A - (A ∩ B)) = 25 - 8 = 17.

  4. Match with options: The result 17 corresponds to option 'd'.

{"correct_answer_key":"d"}

This question is in Thai. Here's the translation and explanation:

Translation:

Given U = {1, 2, 3, ..., 50} and A = {n | n is an even number} B = {n | n is divisible by 3} What is the number of elements in A - (A ∩ B)?

Explanation:

  • A: The set of even numbers in U. A = {2, 4, 6, ..., 50}. The number of elements in A is 50/2 = 25.
  • B: The set of numbers divisible by 3 in U. B = {3, 6, 9, ..., 48}. The number of elements in B is floor(50/3) = 16.
  • A ∩ B: The intersection of A and B, which means the set of numbers that are both even and divisible by 3. These are multiples of 6. A ∩ B = {6, 12, 18, ..., 48}. The number of elements in A ∩ B is floor(50/6) = 8.
  • A - (A ∩ B): This represents the elements in A that are not in A ∩ B. In other words, the even numbers that are not divisible by 3. The number of elements in A - (A ∩ B) is the number of elements in A minus the number of elements in A ∩ B: 25 - 8 = 17.
{"correct_answer_key":"d"}

คำอธิบาย:

  • U = {1, 2, 3, ..., 50} คือเซตของจำนวนเต็ม 1 ถึง 50
  • A = เซตของจำนวนคู่ใน U (คือ 2, 4, 6, ..., 50)
  • B = เซตของจำนวนที่หารด้วย 3 ลงตัวใน U (คือ 3, 6, 9, ..., 48)
  • A ∩ B = เซตของจำนวนที่เป็นทั้งจำนวนคู่และหารด้วย 3 ลงตัว (คือจำนวนที่หารด้วย 6 ลงตัวใน U: 6, 12, 18, 24, 30, 36, 42, 48)
  • A – (A ∩ B) คือจำนวนคู่ที่ไม่หารด้วย 3 ลงตัว

จำนวนสมาชิกของ A:
จำนวนคู่ใน 1 ถึง 50 คือ 2, 4, ..., 50
ใช้สูตรลำดับเลขคณิต:
an = a1 + (n-1)d
50 = 2 + (n-1)×2
48 = (n-1)×2
24 = n-1
n = 25

จำนวนสมาชิกของ A ∩ B:
จำนวนที่หารด้วย 6 ลงตัวใน 1 ถึง 50 คือ 6, 12, 18, 24, 30, 36, 42, 48
ใช้สูตร:
6×k ≤ 50 → k = 1 ถึง 8
ดังนั้นมี 8 ตัว

ดังนั้น A – (A ∩ B) = 25 - 8 = 17

{"correct_answer_key":"d"}

โจทย์กำหนดเซต U = {1, 2, 3, ..., 50}
เซต A = {เลขคู่ใน U}
เซต B = {เลขที่หารด้วย 3 ลงตัวใน U}

ต้องการหาจำนวนสมาชิกของ A – (A ∩ B)
ซึ่งก็คือสมาชิกใน A ที่ไม่อยู่ใน B

  1. จำนวนสมาชิกใน A (เลขคู่ระหว่าง 1 ถึง 50) = 50/2 = 25
  2. สมาชิกใน A ∩ B คือเลขที่เป็นทั้งเลขคู่และหารด้วย 3 ลงตัว
    นั่นคือเลขที่หารด้วย 6 ลงตัว (เพราะ 6 = 2 × 3)
    จำนวนเลขที่หารด้วย 6 ลงตัวระหว่าง 1 ถึง 50 = floor(50/6) = 8
  3. ดังนั้น จำนวนสมาชิกของ A – (A ∩ B) = 25 - 8 = 17

คำตอบคือ 17 ซึ่งตรงกับตัวเลือก d

{"correct_answer_key":"d"}

ให้กำหนด U = {1, 2, 3, ..., 50}
A = {n | n เป็นจำนวนคู่} = คู่จำนวนเต็มในช่วง 1 ถึง 50
B = {n | n หารด้วย 3 ลงตัว} = จำนวนเต็มในช่วง 1 ถึง 50 ที่หารด้วย 3 ลงตัว

  • จำนวนสมาชิกของ A (จำนวนคู่ใน 1 ถึง 50):
    คู่คือ 2, 4, 6, ..., 50
    จำนวนคู่ = 50/2 = 25

  • จำนวนสมาชิกของ B (จำนวนที่หารด้วย 3 ลงตัว):
    3, 6, 9, ..., 48
    จำนวน = 50/3 ≈ 16.66 → จำนวนเต็มคือ 16

  • A ∩ B คือจำนวนคู่ที่หารด้วย 3 ลงตัว:
    คู่และหารด้วย 3 ลงตัว คือ 6, 12, 18, 24, 30, 36, 42, 48
    จำนวน = 8

  • A – (A ∩ B) คือจำนวนคู่ที่ไม่หารด้วย 3 ลงตัว:
    จำนวน = 25 - 8 = 17

ดังนั้น จำนวนสมาชิกของ A – (A ∩ B) เท่ากับ 17

{"correct_answer_key":"d"}

ก่อนอื่นมาวิเคราะห์โจทย์กันก่อน:

  • ( U = {1,2,3,...,50} )
  • ( A = { n \in U \mid n \text{ เป็นจำนวนคู่} } ) หมายถึงจำนวนคู่ตั้งแต่ 1 ถึง 50
  • ( B = { n \in U \mid n \text{ หารด้วย 3 ลงตัว} } ) หมายถึงจำนวนที่หารด้วย 3 ลงตัวตั้งแต่ 1 ถึง 50
  • ( A - (A \cap B) ) หมายถึงจำนวนคู่ที่ไม่หารด้วย 3 ลงตัว

ขั้นตอนการคำนวณ:

  1. จำนวนสมาชิกของเซต ( A ) (จำนวนคู่ตั้งแต่ 1 ถึง 50):

    • มีจำนวนคู่ตั้งแต่ 2,4,6,...,50
    • จำนวนสมาชิก ( = \frac{50}{2} = 25 )

  2. จำนวนสมาชิกของเซต ( A \cap B ) (จำนวนคู่ที่หารด้วย 3 ลงตัว):

    • จำนวนคู่ที่หารด้วย 3 ลงตัว คือจำนวนที่หารด้วย 6 ลงตัว (เพราะต้องหารด้วย 2 และ 3 ลงตัวพร้อมกัน)
    • จำนวนที่หารด้วย 6 ลงตัวตั้งแต่ 1 ถึง 50 คือ 6,12,18,24,30,36,42,48
    • นับได้ทั้งหมด 8 ตัว

  3. ดังนั้น จำนวนสมาชิกของ ( A - (A \cap B) ) คือ:

    • จำนวนสมาชิกของ ( A ) ลบด้วยจำนวนสมาชิกของ ( A \cap B )
    • ( 25 - 8 = 17 )

ดังนั้น คำตอบที่ถูกต้องคือ 17 ซึ่งตรงกับตัวเลือก d

{"correct_answer_key":"d"}

To solve this problem, we need to determine the number of elements in the set ( A - (A \cap B) ).

  1. Set ( U ): This is the universal set containing numbers from 1 to 50.

  2. Set ( A ): This set contains all even numbers from 1 to 50. The even numbers between 1 and 50 are: 2, 4, 6, ..., 50. There are 25 even numbers in this range.

  3. Set ( B ): This set contains numbers divisible by 3 from 1 to 50. The numbers divisible by 3 are: 3, 6, 9, ..., 48. There are 16 such numbers.

  4. Set ( A \cap B ): This set contains numbers that are both even and divisible by 3. These are the numbers divisible by 6 (since 6 is the least common multiple of 2 and 3). The numbers divisible by 6 between 1 and 50 are: 6, 12, 18, ..., 48. There are 8 such numbers.

  5. Set ( A - (A \cap B) ): This set contains elements that are in ( A ) but not in ( A \cap B ). Since ( A ) has 25 elements and ( A \cap B ) has 8 elements, the set ( A - (A \cap B) ) has ( 25 - 8 = 17 ) elements.

Thus, the number of elements in ( A - (A \cap B) ) is 17.

{"correct_answer_key":"d"}

ในการหาจำนวนสมาชิกของ ( A - (A \cap B) ) เราต้องเริ่มจากการหาจำนวนสมาชิกในเซต ( A ) และ ( B ) ก่อน

  1. เซต ( A ): จำนวนคู่ใน ( U ) คือ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50} ซึ่งมีจำนวนสมาชิกทั้งหมด 25 ตัว

  2. เซต ( B ): จำนวนที่หารด้วย 3 ลงตัวใน ( U ) คือ {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48} ซึ่งมีจำนวนสมาชิกทั้งหมด 16 ตัว

  3. เซต ( A \cap B ): จำนวนที่เป็นจำนวนคู่และหารด้วย 3 ลงตัวใน ( U ) คือ {6, 12, 18, 24, 30, 36, 42, 48} ซึ่งมีจำนวนสมาชิกทั้งหมด 8 ตัว

  4. จำนวนสมาชิกของ ( A - (A \cap B) ):

    • จำนวนสมาชิกของ ( A ) คือ 25
    • จำนวนสมาชิกของ ( A \cap B ) คือ 8
    • ดังนั้น ( |A - (A \cap B)| = |A| - |A \cap B| = 25 - 8 = 17 )

ดังนั้นจำนวนสมาชิกของ ( A - (A \cap B) ) เท่ากับ 17 ซึ่งตรงกับตัวเลือก D

{"correct_answer_key":"d"}

Let me break this down step by step. The universal set U is defined as {1, 2, 3, ..., 50}. Set A consists of all even numbers in U, so A = {2, 4, 6, ..., 50}, which has 25 elements (since 50/2 = 25). Set B consists of all numbers in U divisible by 3, so B = {3, 6, 9, ..., 48}, which has 16 elements (since floor(50/3) = 16).

Next, A ∩ B (the intersection of A and B) represents numbers that are both even and divisible by 3, which means numbers divisible by 6 (the least common multiple of 2 and 3). So, A ∩ B = {6, 12, 18, ..., 48}, which has 8 elements (since floor(50/6) = 8).

The problem asks for the number of elements in A – (A ∩ B), which is the set of elements in A that are not in A ∩ B. This is calculated as the number of elements in A minus the number of elements in A ∩ B: 25 – 8 = 17.

Therefore, the correct answer corresponds to option D.

{"correct_answer_key":"d"}

First, the user has provided a JSON object with a question in Thai. I need to understand it and solve it step by step.

The question is:

  • U = {1, 2, 3, ..., 50} (Universal set from 1 to 50)

  • A = {n ∈ U | n is even} (Set of even numbers in U)

  • B = {n ∈ U | n is divisible by 3} (Set of numbers in U divisible by 3)

We need to find the number of elements in A - (A ∩ B), which is the set difference.

A - (A ∩ B) is the same as A minus the intersection of A and B. In other words, it's the elements that are in A but not in both A and B.

More precisely, A - (A ∩ B) = A \ (A ∩ B)

Since A ∩ B is the set of elements in both A and B, A - (A ∩ B) is the elements in A that are not in B.

So, essentially, it's the even numbers in U that are not divisible by 3.

First, let's find the sets:

  • A: Even numbers from 1 to 50. So, 2, 4, 6, ..., 50. Number of even numbers: 50/2 = 25.

  • B: Numbers divisible by 3 from 1 to 50. So, 3, 6, 9, ..., 48. (50/3 = 16.666, so up to 48) Number: floor(50/3) = 16, but wait: first is 31=3, last is 316=48, so 16 elements.

  • A ∩ B: Numbers that are even and divisible by 3, which means divisible by LCM of 2 and 3, which is 6. So, multiples of 6 from 1 to 50: 6, 12, 18, ..., 48. First: 61=6, last: 68=48 (since 6*9=54>50), so 8 elements.

Now, A - (A ∩ B) = A minus the elements that are in A ∩ B.

So, number of elements in A is 25, number in A ∩ B is 8, so A - (A ∩ B) has 25 - 8 = 17 elements.

Therefore, the answer should be 17.

Looking at the options: a:"8", b:"9", c:"15", d:"17", e:"19"

So, d:"17"

I need to respond as per the initial instruction: Provide a brief explanation, then respond in JSON with the "correct_answer_key".

First, brief explanation: Summarize the reasoning.

Then, output: {"correct_answer_key":"d"}

Make sure the explanation is brief.

Explanation: U is {1 to 50}. A is even numbers: 25 elements. B is multiples of 3: not directly needed except for intersection. A ∩ B is multiples of 6: 8 elements. So, A minus A∩B has 25 - 8 = 17 elements.

Yes.

Brief Explanation:

กำหนดให้ U = {1, 2, ..., 50}

  • A คือเซตของจำนวนคู่ใน U: มีสมาชิก 25 ตัว (2, 4, ..., 50)
  • B คือเซตของจำนวนที่หารด้วย 3 ลงตัวใน U
  • A ∩ B คือเซตของจำนวนที่หารด้วย 6 ลงตัว (LCM ของ 2 และ 3): มีสมาชิก 8 ตัว (6, 12, ..., 48)
  • A - (A ∩ B) คือสมาชิกของ A ที่ไม่อยู่ใน A ∩ B: จึงมี 25 - 8 = 17 สมาชิก

{"correct_answer_key":"d"}

ก่อนอื่นเราต้องกำหนดชุด A และ B ก่อน

A = {2, 4, 6, ..., 50} (จำนวนคู่ตั้งแต่ 1 ถึง 50) B = {3, 6, 9, ..., 48} (จำนวนเต็มที่หารด้วย 3 ลงตัวตั้งแต่ 1 ถึง 50)

A ∩ B = {6, 12, 18, ..., 48} (จำนวนคู่ที่หารด้วย 3 ลงตัว)

จำนวนสมาชิกของ A คือ 25 (มี 25 จำนวนคู่ตั้งแต่ 1 ถึง 50) จำนวนสมาชิกของ A ∩ B คือ 8 (มี 8 จำนวนคู่ที่หารด้วย 3 ลงตัวตั้งแต่ 1 ถึง 50)

ดังนั้น จำนวนสมาชิกของ A - (A ∩ B) คือ 25 - 8 = 17

{"correct_answer_key":"d"}

To solve this problem, we first need to understand the given sets.

  • U is the universal set containing integers from 1 to 50.
  • A is the set of elements in U such that n is even. So, A = {2, 4, 6, ..., 50}.
  • B is the set of elements in U such that n is divisible by 3. So, B = {3, 6, 9, ..., 48}.

The intersection of A and B (A ∩ B) contains elements that are both even and divisible by 3, meaning they are divisible by 6. So, A ∩ B = {6, 12, 18, ..., 48}.

To find the number of elements in A ∩ B, we notice it's an arithmetic sequence with the first term a = 6 and the common difference d = 6. The last term is 48. We can find the number of terms (n) using the formula for the nth term: a + (n-1)d = 48. 6 + (n-1)6 = 48 (n-1)6 = 42 n-1 = 7 n = 8

So, there are 8 elements in A ∩ B.

The set A has 25 elements since there are 25 even numbers between 1 and 50.

The expression A - (A ∩ B) means we are removing the elements that are in both A and B from A. So, we are left with elements that are in A but not in B.

The number of elements in A - (A ∩ B) = Number of elements in A - Number of elements in A ∩ B = 25 - 8 = 17

Therefore, the number of elements in A - (A ∩ B) is 17.

{"correct_answer_key":"d"}

Step 1: Identify the sets U, A, and B

The universal set U contains all natural numbers from 1 to 50.

Step 2: Determine set A

Set A consists of even numbers (natural numbers divisible by 2) from 1 to 50. So, A = {2, 4, 6, ..., 50}. There are 25 even numbers from 1 to 50.

Step 3: Determine set B

Set B consists of natural numbers divisible by 3 from 1 to 50. So, B = {3, 6, 9, ..., 48}.

Step 4: Find the intersection of A and B (A ∩ B)

To find A ∩ B, we need to identify numbers that are divisible by both 2 and 3, which means divisible by 6. The numbers are: 6, 12, 18, 24, 30, 36, 42, 48.

Step 5: Calculate the number of elements in set A and set B

Set A has 25 elements (even numbers from 2 to 50), and set B has 16 elements (numbers divisible by 3 from 1 to 50, but we focus on the count for intersection).

Step 6: Find A ∩ B (numbers divisible by both 2 and 3)

Numbers divisible by 6 from 1 to 50 are: 6, 12, 18, 24, 30, 36, 42, 48. There are 8 numbers.

Step 7: Calculate A - B

A - B refers to elements in A that are not in B. Since A has 25 elements and 8 of them are also in B (as multiples of 6), we subtract these 8 from A's total to find elements only in A.

Step 8: Identify the correct answer

The question seems to be asking for the number of elements in A that are not in B (or not divisible by 3), but given the context, it seems there might be confusion in interpreting "A – (A^B)".

Step 9: Correct Interpretation

Given the likely intended meaning is to find elements in A not divisible by 3 (or elements in A not in B), let's correct course:

  • Set A has 25 elements (all even numbers).
  • Set B has numbers divisible by 3.

Step 10: Elements in A but not in B (or multiples of 3)

Elements in A (even numbers) but not in B (multiples of 3) would be those even numbers not divisible by 3.

Step 11: Calculate elements in A not in B

Elements in A but not in B (or not multiples of 3) would be those even numbers not divisible by 3. The even numbers not divisible by 3 from 1 to 50 are: 2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50.

Step 12: Count the elements

There are 17 numbers that fit this criterion (not divisible by 3 but are even).

Step 13: Provide the correct answer in JSON format

The question seems to relate to finding elements in a set that are not divisible by 3, hence focusing on the count of numbers that are even but not multiples of 3.

The final answer is:

{"correct_answer_key":"c"}

To solve this problem, we need to determine the number of elements in the set ( A - (A \cap B) ), where:

  • ( U = {1, 2, 3, \ldots, 50} )
  • ( A = {n \in U \mid n \text{ is even}} )
  • ( B = {n \in U \mid n \text{ is divisible by 3}} )

First, let's find the elements of sets ( A ) and ( B ):

  • ( A = {2, 4, 6, \ldots, 50} )
  • ( B = {3, 6, 9, \ldots, 48} )

Next, we find the intersection ( A \cap B ):

  • ( A \cap B = {6, 12, 18, 24, 30, 36, 42, 48} )

There are 8 elements in ( A \cap B ).

Now, we need to find the number of elements in ( A - (A \cap B) ):

  • ( A ) has 25 elements (all even numbers from 2 to 50).
  • ( A \cap B ) has 8 elements.

Thus, the number of elements in ( A - (A \cap B) ) is:

[ 25 - 8 = 17 ]

Therefore, the correct answer is option D.

{"correct_answer_key":"d"}

อธิบายสั้น ๆ:
• A เป็นเซตของจำนวนคู่ใน U (2, 4, 6, 8, …, 50) มีสมาชิกทั้งหมด 25 ตัว
• B เป็นเซตของจำนวนที่หารด้วย 3 ลงตัวใน U (3, 6, 9, …, 48)
• A ∩ B จึงเป็นเซตของจำนวนที่เป็นคู่และหาร 3 ลงตัว หรือก็คือ “จำนวนที่หารด้วย 6 ลงตัว” ในช่วง 1 ถึง 50 (6, 12, 18, 24, 30, 36, 42, 48) มีสมาชิกทั้งหมด 8 ตัว
• A – (A ∩ B) จึงมีจำนวนสมาชิกเท่ากับ 25 – 8 = 17

ดังนั้นคำตอบคือข้อ d

{"correct_answer_key":"d"}

เซต A มีตัวเลขคู่ในช่วง 1‑50 จึงมี 50 ÷ 2 = 25 สมาชิก
เซต A∩B คือจำนวนที่เป็นทั้งคู่และหาร 3 ลงตัว ⇒ ต้องเป็นพหุคูณของ 6 ภายใน 1‑50
พหุคูณของ 6 ได้แก่ 6, 12, 18, 24, 30, 36, 42, 48 รวม 8 ตัว

จำนวนสมาชิกของ A – (A∩B) = 25 – 8 = 17

{"correct_answer_key":"d"}

เซต A มีจำนวนคู่ภายใน 1‑50 ทั้งหมด 25 ตัว (2,4,…,50)
เซต A ∩ B คือจำนวนที่เป็นทั้งคู่และหาร 3 ลงตัว → เป็นพหุคูณของ 6 ≤ 50 ได้แก่ 8 ตัว (6,12,18,24,30,36,42,48)
ดังนั้น |A – (A ∩ B)| = 25 – 8 = 17

{"correct_answer_key":"d"}

เซต A มีเลขคู่ในช่วง 1‑50 ทั้งหมด 25 ตัว
เซต A ∩ B คือเลขที่หารด้วย 6 ลงตัว (6, 12, … , 48) มี 8 ตัว
A – (A ∩ B) จึงเหลือ 25 – 8 = 17 ตัว

{"correct_answer_key":"d"}

ในเซต U = {1, 2, 3, ..., 50}
• เซต A คือ จำนวนคู่ระหว่าง 1 ถึง 50 จำนวนสมาชิกของ A มีทั้งหมด 25 จำนวน (2, 4, 6, ..., 50)
• เซต B คือ จำนวนที่หารด้วย 3 ลงตัว
• ส่วน A ∩ B คือจำนวนที่เป็นทั้งคู่และหารด้วย 3 ลงตัว ซึ่งจะได้เป็นจำนวนที่หารด้วย 6 ลงตัว: 6, 12, 18, 24, 30, 36, 42, 48 จำนวนสมาชิกมี 8 ตัว
ดังนั้น A – (A ∩ B) จะได้จำนวนสมาชิก = 25 - 8 = 17

{"correct_answer_key": "d"}

A คือเซตของจำนวนคู่ใน U โดยมีสมาชิกทั้งหมด 25 ตัว (2, 4, 6, ..., 50) ส่วน A^B หรือ A ∩ B คือจำนวนคู่ที่หารด้วย 3 ลงตัว ซึ่งหมายถึงจำนวนที่หารด้วย 6 ลงตัวใน U ได้แก่ 6, 12, 18, 24, 30, 36, 42, 48 จำนวน 8 ตัว ดังนั้น สมาชิกของ A – (A^B) จะได้ 25 - 8 = 17

{"correct_answer_key": "d"}

We start with set U = {1,2,3,...,50}. Set A contains all even numbers between 1 and 50, so there are 25 even numbers. Set B contains numbers divisible by 3. The intersection A ∩ B consists of numbers that are both even and divisible by 3, i.e., divisible by 6. There are ⌊50/6⌋ = 8 such numbers. Therefore, A - (A ∩ B) has 25 - 8 = 17 elements.

{"correct_answer_key": "d"}

เซต A มีจำนวนคู่ใน 1–50 ทั้งหมด 25 ตัว (2,4,…,50)
A ∩ B คือจำนวนที่เป็นทั้งคู่และหารด้วย 3 ลงตัว (เท่ากับหารด้วย 6) มี ⌊50/6⌋ = 8 ตัว
ดังนั้น |A – (A ∩ B)| = 25 – 8 = 17

{"correct_answer_key":"d"}

A ประกอบด้วยเลขคู่ 1–50 มี 25 ตัว (2,4,…,50)
A ∩ B คือเลขที่เป็นทั้งคู่และหาร 3 ลงตัว = เลขที่หาร 6 ลงตัว มี 8 ตัว (6,12,…,48)
ดังนั้น |A – (A ∩ B)| = 25 – 8 = 17

{"correct_answer_key":"d"}

A ประกอบด้วยจำนวนคู่ 2–50 มี 25 ตัวเลข ส่วนที่ทั้งคู่และหาร 3 ลงตัวคือจำนวนที่เป็น 6’s จาก 6 ถึง 48 มี 8 ตัว จึงเหลือ 25–8 = 17

{"correct_answer_key":"d"}

ในการแก้ปัญหานี้ เราจำเป็นต้องหาจำนวนสมาชิกของ ( A - (A \cap B) ) โดยที่:

  • ( U = {1, 2, 3, \ldots, 50} )
  • ( A = {n \mid n \text{ เป็นจำนวนคู่}} )
  • ( B = {n \mid n \text{ หารด้วย 3 ลงตัว}} )

ขั้นตอนการคำนวณ:

  1. หาจำนวนสมาชิกของ ( A ):

    • จำนวนคู่ในชุด ( U ) คือ ( 2, 4, 6, \ldots, 50 )
    • นี่เป็นลำดับเรขาคณิตที่มี ( a_1 = 2 ), ( d = 2 ), และ ( a_n = 50 )
    • ใช้สูตร ( a_n = a_1 + (n-1)d ) เพื่อหาจำนวนเทอม ( n ): [ 50 = 2 + (n-1) \times 2 \ 48 = 2(n-1) \ 24 = n-1 \ n = 25 ]
    • ดังนั้น ( |A| = 25 )

  2. หาจำนวนสมาชิกของ ( B ):

    • จำนวนที่หารด้วย 3 ลงตัวในชุด ( U ) คือ ( 3, 6, 9, \ldots, 48 )
    • นี่เป็นลำดับเรขาคณิตที่มี ( a_1 = 3 ), ( d = 3 ), และ ( a_n = 48 )
    • ใช้สูตร ( a_n = a_1 + (n-1)d ) เพื่อหาจำนวนเทอม ( n ): [ 48 = 3 + (n-1) \times 3 \ 45 = 3(n-1) \ 15 = n-1 \ n = 16 ]
    • ดังนั้น ( |B| = 16 )

  3. หาจำนวนสมาชิกของ ( A \cap B ):

    • จำนวนที่เป็นคู่และหารด้วย 3 ลงตัวคือจำนวนที่หารด้วย 6 ลงตัว
    • จำนวนที่หารด้วย 6 ลงตัวในชุด ( U ) คือ ( 6, 12, 18, \ldots, 48 )
    • นี่เป็นลำดับเรขาคณิตที่มี ( a_1 = 6 ), ( d = 6 ), และ ( a_n = 48 )
    • ใช้สูตร ( a_n = a_1 + (n-1)d ) เพื่อหาจำนวนเทอม ( n ): [ 48 = 6 + (n-1) \times 6 \ 42 = 6(n-1) \ 7 = n-1 \ n = 8 ]
    • ดังนั้น ( |A \cap B| = 8 )

  4. หาจำนวนสมาชิกของ ( A - (A \cap B) ):

    • ( A - (A \cap B) ) คือจำนวนที่อยู่ใน ( A ) แต่ไม่อยู่ใน ( A \cap B )
    • ดังนั้นจำนวนสมาชิกคือ ( |A| - |A \cap B| = 25 - 8 = 17 )

ดังนั้น จำนวนสมาชิกของ ( A - (A \cap B) ) คือ 17 ซึ่งตอบคำถามในตัวเลือก ( d ).

{"correct_answer_key":"d"}

Explanation:

We are tasked with determining the number of elements in the set $ A - (A \cap B) $, where:

  • $ U = {1, 2, 3, \dots, 50} $ is the universal set.
  • $ A = {n \in U \mid n \text{ is even}} $, i.e., $ A $ contains all even numbers in $ U $.
  • $ B = {n \in U \mid n \text{ is divisible by 3}} $, i.e., $ B $ contains all numbers divisible by 3 in $ U $.

Step 1: Analyze $ A $

The even numbers in $ U $ are: $$ A = {2, 4, 6, 8, \dots, 50}. $$ This is an arithmetic sequence with:

  • First term $ a = 2 $,
  • Common difference $ d = 2 $,
  • Last term $ l = 50 $.

The number of terms in $ A $ is given by: $$ n = \frac{l - a}{d} + 1 = \frac{50 - 2}{2} + 1 = 25. $$ Thus, $ |A| = 25 $.

Step 2: Analyze $ A \cap B $

The set $ A \cap B $ contains numbers that are both even and divisible by 3, i.e., numbers divisible by $ \text{lcm}(2, 3) = 6 $. These numbers are: $$ A \cap B = {6, 12, 18, \dots, 48}. $$ This is an arithmetic sequence with:

  • First term $ a = 6 $,
  • Common difference $ d = 6 $,
  • Last term $ l = 48 $.

The number of terms in $ A \cap B $ is: $$ n = \frac{l - a}{d} + 1 = \frac{48 - 6}{6} + 1 = 8. $$ Thus, $ |A \cap B| = 8 $.

Step 3: Compute $ A - (A \cap B) $

The set $ A - (A \cap B) $ contains all elements in $ A $ that are not in $ A \cap B $. Its size is: $$ |A - (A \cap B)| = |A| - |A \cap B| = 25 - 8 = 17. $$

Thus, the number of elements in $ A - (A \cap B) $ is $ 17 $, which corresponds to option $ \text{d} $.

Final Answer:

{"correct_answer_key":"d"}

Okay, let's tackle this problem step by step. The question is in Thai, so I need to make sure I understand all the terms. The universal set U is {1,2,3,...,50}. Set A is defined as {neu | n เป็นจำนวนคู่}, which translates to "neu where n is even". Wait, "neu" might be a typo or maybe "n ที่เป็นจำนวนคู่" meaning "n that are even numbers". So A is the set of even numbers in U.

Set B is {neu | n หารด้วย 3 ลงตัว}, which means "n divisible by 3". So B is the set of multiples of 3 in U. The question asks for the number of elements in A minus (A intersect B).

First, let me rephrase the problem in my own words. We need to find |A - (A ∩ B)|. Wait, actually, the notation might be A minus (A ∩ B). Alternatively, maybe it's A complement? But the question says A – (A^B). The caret symbol here might represent intersection, since sometimes in set theory, especially in some notations, A^B could be A intersect B. So A minus (A intersect B) would be elements in A that are not in B.

Alternatively, maybe the question is written as A minus the intersection of A and B. Let me confirm. The operation A - (A ∩ B) would indeed be all elements in A that are not in B. Because A ∩ B is the elements common to both A and B, so subtracting that from A leaves the elements in A that are not in B.

So essentially, the problem is asking for the number of even numbers in U that are not divisible by 3.

First, let's compute the number of elements in A. Since U is from 1 to 50, the even numbers are 2,4,6,...,50. The count of these is 50 divided by 2, so 25 elements. So |A| =25.

Next, we need to find the number of elements in A ∩ B. This is the set of numbers that are both even and divisible by 3. A number divisible by both 2 and 3 must be divisible by 6. So A ∩ B is the set of multiples of 6 in U.

To find how many multiples of 6 are there between 1 and 50: the first multiple is 6, then 12, 18,... up to the largest multiple less than or equal to 50. Let's compute 50 divided by 6, which is approximately 8.333. So the integer part is 8, so there are 8 multiples of 6 up to 48 (since 6*8=48). The next multiple would be 54, which is over 50. So |A ∩ B| =8.

Therefore, the number of elements in A - (A ∩ B) is |A| - |A ∩ B| =25 -8=17. Wait, but looking at the options given, the choices are a:8, b:9, c:15, d:17, e:19. So 17 is option d.

Wait, but let me double-check my calculations to be sure.

First, count of even numbers between 1-50: 50/2=25. That's correct.

Multiples of 6 up to 50: Let's list them. 6,12,18,24,30,36,42,48. That's 8 numbers. So 8 elements.

25-8=17. So the answer should be 17, which is option d.

Hmm, but let me check if there's any mistake here. For example, maybe I made an error in counting the multiples of 6. Let me count again:

Starting at 61=6, 62=12, 63=18, 64=24, 65=30, 66=36, 67=42, 68=48, 6*9=54. 54 is over 50, so up to 8. So 8 elements. That's correct.

So 25-8=17. So the correct answer is d)17.

Alternatively, maybe I misunderstood the question. Let me check the original problem again.

The question says "จำนวนสมาชิกของ A – (A^B) เท่ากับเท่าใด". The symbol ^ might not be intersection. In some contexts, A^B could be symmetric difference, but usually that's denoted by a triangle or AΔB. Alternatively, maybe it's a typo and they meant intersection. Since in Thai set notation, sometimes they use ^ for intersection? Not sure, but given the context, the most logical interpretation is intersection.

Alternatively, if A^B is the union, then A - (A ∪ B) would be elements in A not in B, but that's the same as A - B, which is the same as A minus the intersection. Wait no, A - (A ∪ B) would be elements in A not in A ∪ B, but since A is a subset of A ∪ B, that would be zero. So that can't be.

Alternatively, maybe the question is written as A complement minus B? But that's probably not.

Alternatively, maybe the question is A minus B, which is the same as A - (A ∩ B). Because A - B is equivalent to A minus the intersection. So yes, the calculation is correct.

Therefore, the answer should be d)17.

The problem asks for the number of elements in set ( A - (A \cap B) ). Here's the breakdown:

  1. Set ( A ): Contains even numbers from 1 to 50. There are ( \frac{50}{2} = 25 ) elements.
  2. Set ( A \cap B ): Numbers divisible by both 2 and 3 (i.e., multiples of 6). The multiples of 6 up to 50 are ( 6, 12, 18, 24, 30, 36, 42, 48 ), totaling 8 elements.
  3. Result: Subtract ( |A \cap B| ) from ( |A| ):
    ( 25 - 8 = 17 ).

Thus, the correct answer is d) 17.

{"correct_answer_key":"d"}

เพื่อหาจำนวนสมาชิกของ A - (A ∩ B) เราต้องหาจำนวนสมาชิกของ A และ A ∩ B ก่อน

A คือชุดของจำนวนคู่ใน U = {1, 2, 3, ..., 50} ดังนั้น A = {2, 4, 6, ..., 50} ซึ่งมีจำนวนสมาชิก 25 (เนื่องจากมีจำนวนคู่ 25 ตัวในช่วง 1 ถึง 50)

B คือชุดของจำนวนใน U ที่หารด้วย 3 ลงตัว ดังนั้น B = {3, 6, 9, ..., 48} ซึ่งมีจำนวนสมาชิก 16 (เนื่องจากมีจำนวน 16 ตัวในช่วง 1 ถึง 50 ที่หารด้วย 3 ลงตัว)

A ∩ B คือชุดของจำนวนคู่ที่หารด้วย 3 ลงตัว ดังนั้น A ∩ B = {6, 12, 18, ..., 48} ซึ่งมีจำนวนสมาชิก 8 (เนื่องจากมีจำนวน 8 ตัวในช่วง 1 ถึง 50 ที่เป็นจำนวนคู่และหารด้วย 3 ลงตัว)

ดังนั้น จำนวนสมาชิกของ A - (A ∩ B) คือจำนวนสมาชิกของ A ลบด้วยจำนวนสมาชิกของ A ∩ B ซึ่งคือ 25 - 8 = 17

{"correct_answer_key":"d"}

Alright! Let's tackle this problem step by step. The user provided a set theory question in Thai, so I need to make sure I understand it correctly. The universal set U is {1,2,3,...,50}. Set A consists of even numbers in U, and set B consists of numbers divisible by 3 in U. The question asks for the number of elements in A minus the intersection of A and B, denoted as A – (A ∩ B).

First, I need to find the number of elements in set A. Since A contains all even numbers from 1 to 50, that would be every second number. So, starting from 2 up to 50, there are 25 even numbers. Therefore, |A| = 25.

Next, I need to determine the number of elements in the intersection of A and B, which is A ∩ B. This set includes numbers that are both even and divisible by 3. In other words, these are numbers divisible by 6 because 6 is the least common multiple of 2 and 3. From 1 to 50, the multiples of 6 are 6, 12, 18, ..., up to 48. To find how many there are, I can divide 48 by 6, which gives 8. So, |A ∩ B| = 8.

Now, to find the number of elements in A minus the intersection of A and B, I subtract the size of the intersection from the size of A. That is, |A| - |A ∩ B| = 25 - 8 = 17.

Looking at the answer choices, option d is 17. Therefore, the correct answer should be option d.

{
  "correct_answer_key": "d"
}