Question

Three sets A, B and C are defined as follows:
\(A=\{x|x\text{ is an even integer}\}\),
\(B=\{x|x\text{ is an integer and multiple of 5}\}\),
\(C=\{x|x\text{ is an integer and multiple of 10}\}\).
Then \(A \cap (B \cup C)=\)

• A. \(\{x|x\text{ is a multiple of 30}\}\)
• B. \(\{x|x\text{ is a multiple of 20}\}\)
• C. \(\{x|x\text{ is a multiple of 15}\}\)
• D. \(\{x|x\text{ is a multiple of 10}\}\)

Hint:

Whenever one set is completely contained in another, their union equals the larger set. Here \(C\subseteq B\), so \(B\cup C=B\).

Answer 1

The Correct Option is D

Concept: The union of two sets contains all elements belonging to either set, while the intersection contains only those elements common to both sets.

Step 1: Simplify \(B\cup C\).
Set \(B\) contains all multiples of 5 and set \(C\) contains all multiples of 10. Since every multiple of 10 is automatically a multiple of 5, \[ C\subseteq B \] Therefore, \[ B\cup C=B \]

Step 2: Find \(A\cap B\).
Now, \[ A\cap(B\cup C)=A\cap B \] Set \(A\) contains all even integers and set \(B\) contains all multiples of 5. A number that is both even and a multiple of 5 must be divisible by \[ \operatorname{LCM}(2,5)=10 \] Hence, \[ A\cap B=\{x|x\text{ is a multiple of }10\} \]