Question

If $A=(5,6,7,8,9),B=\{x:3<x<8\text{and}x\in W\}$ and $C=\{x:x\le 5\text{and}x\in N\}$. Is (A $\cup$ B) $\cup$ C = A $\cup$ (B $\cup$ C) ? (Type $1$ if true and $0$ if false)

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 $5$.
Union of two sets has all the elements of both the sets.
So, $A\cup B=$ { $4,5,6,7,8,9$ }
And, $(A\cup B)\cup C=$ { $4,5,6,7,8,9$ }$\cup$ { $1,2,3,4,5$ } $=$ { $1,2,3,4,5,6,7,8,9$ } ....(1)
Also, $B\cup C=$ { $1,2,3,4,5,6,7$ }
And, $A\cup (B\cup C)=$ { $5,6,7,8,9$ }$\cup$ { $1,2,3,4,5,6,7$ } $=$ { $1,2,3,4,5,6,7,8,9$ } ....(2)
From (1) and (2), we have
$(A\cup B)\cup C=A\cup (B\cup C)$