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Step-by-Step Solution
Step 1: Identify the Set X
We are given
X = \{ n \in \mathbb{N} : 1 \le n \le 50 \}.
This means X contains all natural numbers from 1 to 50.
Step 2: Define the Sets A and B
• A = \{ n \in X : n \text{ is a multiple of } 2\}
• B = \{ n \in X : n \text{ is a multiple of } 7\}
Step 3: Count the Number of Elements in A
Multiples of 2 from 1 to 50 are 2, 4, 6, …, 50. The largest even number not exceeding 50 is 50 itself, so these are exactly the even numbers between 1 and 50.
There are 25 even numbers from 2 to 50 (since 50/2 = 25). Thus, n(A) = 25 .
Step 4: Count the Number of Elements in B
Multiples of 7 from 1 to 50 are 7, 14, 21, 28, 35, 42, 49.
Thus, n(B) = 7 .
Step 5: Determine A ∩ B
A ∩ B consists of the common elements that are multiples of both 2 and 7. Such numbers are multiples of 14, since the least common multiple of 2 and 7 is 14.
So within 1 to 50, the multiples of 14 are 14, 28, 42.
Therefore, n(A \cap B) = 3 .
Step 6: Use the Formula for the Union
The number of elements in A \cup B is given by:
n(A \cup B) = n(A) + n(B) - n(A \cap B).
Substituting the values:
n(A \cup B) = 25 + 7 - 3 = 29.
Final Answer
The number of elements in the smallest subset of X containing both A and B is 29.
