Question

Given P = {a, b, c, d, e}, Q = {a, e, i, o, u} and R = {a, c, e, g}, verify associative property of union and intersection of sets.

Answer 1

We know that two sets $A$ and $B$ are associative if $A\cap (B\cap C)=(A\cap B)\cap C$ and $A\cup (B\cup C)=(A\cup B)\cup C$.

The given sets are $P=${$a,b,c,d,e$}, $Q=${$a,e,i,o,u$} and $R=${$a,c,e,g$}

(i) $P\cap (Q\cap R)=(P\cap Q)\cap R$

$P\cap (Q\cap R)=\{a,b,c,d,e\}\cap \left[\right\{a,e,i,o,u\}\cap \{a,c,e,g\left\}\right]=\{a,b,c,d,e\}\cap \{a,e\}=\{a,e\}......\left(1\right)(P\cap Q)\cap R=\left[\right\{a,b,c,d,e\}\cap \{a,e,i,o,u\left\}\right]\cap \{a,c,e,g\}=\{a,e\}\cap \{a,c,e,g\}=\{a,e\}......\left(2\right)$

From equations 1 and 2, we get that $P\cap (Q\cap R)=(P\cap Q)\cap R$.

(ii) $P\cup (Q\cup R)=(P\cup Q)\cup R$

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

$\{a,b,c,d,e\}\cup \{a,c,e,q,i,o,u\}=\{a,b,c,d,e,g,i,o,u\}......\left(3\right)(P\cup Q)\cup R=\left[\right\{a,b,c,d,e\}\cup \{a,e,i,o,u\left\}\right]\cup \{a,c,e,g\}=$

$\{a,b,c,d,e,i,o,u\}\cup \{a,c,e,g\}=\{a,b,c,d,e,g,i,o,u\}......\left(4\right)$

From equations 3 and 4, we get that $P\cup (Q\cup R)=(P\cup Q)\cup R$.

Hence, the sets $P=${$a,b,c,d,e$}, $Q=${$a,e,i,o,u$} and $R=${$a,c,e,g$} are associative.