Question

If $A=(5,6,7,8,9),B=\{x:3<x<8\text{and}x\in W\}$ and $C=\{x:x\le 5\text{and}x\in N\}$, then find : (A $\cap$ B) $\cap$ C - A $\cap$ (B $\cap$ C) $=?$

Answer 1

From the given sets, we have

$B$ $=$$\{$ $4,5,6,7$ $\}$ as $x$ is greater than $3$ but less than $8$.

And $C$ $=$ $\{$ $1,2,3,4,5$ $\}$ as $x$ is a natural number less than equal to $5$.

Intersection of two sets has the elements common in both the sets

So, $A\cap B=$ $\{$ $5,6,7$ $\}$

And, $(A\cap B)\cap C=$ $\{$ $5,6,7$ $\}$$\cap$ $\{$ $1,2,3,4,5$ $\}$ $=$ $\{$ $5$ $\}$ ....(1)

Also, $B\cap C=$ $\{$ $4,5$ $\}$

And, $A\cap (B\cap C)=$ $\{$ $5,6,7,8,9$ $\}$$\cap$ $\{$ $4,5$ $\}$ $=$ $\{$ $5$ $\}$ ....(2)

From (1) and (2), we know

$(A\cap B)\cap C=A\cap (B\cap C)$