If A={1,2,3}, B={2,4,6,8,10}, and C={1,5,9}, which of the following statements is true?

A. C∩(A∩B)={1,2,5,9}

B. C∪(A∪B)={1,2,5,9} 

C. C∪(A∩B)={1,2,5,9}

D. C∩(A∪B)={1,2,5,9}

As a reminder, $\cap$ represents the intersection of the sets, and will output the common terms in the sets being compared.

$\cup$ represents the union of the sets, and will combine all terms written in the sets being compared into one set.

Following the order of operations, we must perform the operation written in the parenthesis first for each choice.

 $$\begin{gathered} C\cap (A\cap B)=\{1,5,9\}\cap \left(\right\{1,2,3)\cap \{2,4,6,8,10\left\}\right) \\ C\cap (A\cap B)=\{1,5,9\}\cap \left\{2\right\} \\ C\cap (A\cap B)=\{ \} \\ Therefore, A is false \end{gathered}$$

$$\begin{gathered} C\cup (A\cup B)=\{1,5,9\}\cup \left(\right\{1,2,3)\cup \{2,4,6,8,10\left\}\right) \\ C\cup (A\cup B)=\{1,5,9\}\cup \{1,2,3,4,6,8,10\} \\ C\cup (A\cup B)=\{1,2,3,4,5,6,8,9,10\} \\ Therefore, B is false \end{gathered}$$

$$\begin{gathered} C\cup (A\cap B)=\{1,5,9\}\cup \left(\right\{1,2,3)\cap \{2,4,6,8,10\left\}\right) \\ C\cup (A\cap B)=\{1,5,9\}\cup \left\{2\right\} \\ C\cup (A\cap B)=\{1,2,5,9\} \\ Therefore, C is true \end{gathered}$$

$$\begin{gathered} C\cap (A\cup B)=\{1,5,9\}\cap \left(\right\{1,2,3)\cup \{2,4,6,8,10\left\}\right) \\ C\cap (A\cup B)=\{1,5,9\}\cap \{1,2,3,4,6,8,10\} \\ C\cap (A\cup B)=\left\{1\right\} \\ Therefore, D is false \end{gathered}$$

From above, C is our only true statement and is our answer.