Given $A=\{2,4,6,8\}$, $B=\{4,8,12,16\}$, $C=\left\{8\right\}$, find the following:
(C) $C\cap \varphi$

• A. $\left\lbrace8\right\rbrace$
• B. $\left\lbrace\right\rbrace$
• C. $\left\lbrace0\right\rbrace$

Answer: (B)

Verified step by step guidance

1. Recall that the intersection of two sets, say \(X\) and \(Y\), denoted by \(X \cap Y\), is the set of all elements that are common to both \(X\) and \(Y\).
2. Identify the sets given: \(C = \left\lbrace 8 \right\rbrace\) and \(\phi\) (the empty set) which contains no elements.
3. Since \(\phi\) has no elements, there are no elements that can be common between \(C\) and \(\phi\).
4. Therefore, the intersection \(C \cap \phi\) is the set containing all elements common to both, which in this case is no elements.
5. Express the result as the empty set: \(C \cap \phi = \left\lbrace \right\rbrace\).