Question

For A={x|x is a prime factor of 24}, B=(x|5 < x $\le$ 12, x $ϵ$ N) and C= { 1, 4, 5, 6}

Verify $A\cup (B\cup C)=(A\cup B)\cup C$.

Answer 1

The given sets are $A=\{2,3\}$ because $x$ is a prime factor of $24$, $B=\{6,7,8,9,10,11,12\}$ because $5<x\le 12$ and $C=\{1,4,5,6\}$

Let us first find $A\cup (B\cup C)$ as follows:

$A\cup (B\cup C)=\{2,3\}\cup (\{6,7,8,9,10,11,12\}\cup \{1,4,5,6\})=\{2,3\}\cup \{1,4,5,6,7,8,9,10,11,12\}$

$=\{1,2,3,4,5,6,7,8,9,10,11,12\}........\left(1\right)$

Now we find $(A\cup B)\cup C$as follows:

$(A\cup B)\cup C=(\{2,3\}\cup \{6,7,8,9,10,11,12\})\cup \{1,4,5,6\}=\{2,3,6,7,8,9,10,11,12\}\cup \{1,4,5,6\}$

$=\{1,2,3,4,5,6,7,8,9,10,11,12\}........\left(2\right)$

Since equation 1 is equal to equation 2,

Hence, $A\cup (B\cup C)=(A\cup B)\cup C$