Question

Given P={ a, b, c, d, e}, Q={ a, e, i, o, u} and R={ a, c, e, g}. Verify the associative property of set intersection.

Answer 1

Three sets $A,B$ and $C$ are said to be associative if $A\cap (B\cap C)=(A\cap B)\cap C$.

Here, the given sets are $P=\{a,b,c,d,e\}$, $Q=\{a,e,i,o,u\}$ and $C=\{a,c,e,g\}$

Let us first find $P\cap (Q\cap R)$ as follows:

$P\cap (Q\cap R)=\{a,b,c,d,e\}\cap (\{a,e,i,o,u\}\cap \{a,c,e,g\})=\{a,b,c,d,e\}\cap \{a,e\}=\{a,e\}........\left(1\right)$

Now we find $(P\cap Q)\cap R$as follows:

$(P\cap Q)\cap R=(\{a,b,c,d,e\}\cap \{a,e,i,o,u\})\cap \{a,c,e,g\}=\{a,e\}\cup \{a,c,e,g\}=\{a,e\}........\left(2\right)$

Since equation 1 is equal to equation 2, therefore, $P\cap (Q\cap R)=(P\cap Q)\cap R$

Hence, the sets $P,Q$ and $R$ satisfies the associative property of intersection.