Question

Given $A=$ {$x\in N$ : $x<6$}, $B=$ {$3,6,9$} and $C=$ {$x\in N$ : $2x-5\le 8$}. Show that :

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

Answer 1

$A=$ { $1,2,3,4,5$ } as $x$ is a natural number less than $6$
For C,
$2x-5\le 8$
$=>2x\le 13$
$=>x\le 6.5$
So, $C=$ { $1,2,3,4,5$ } as $x$ is a natural number less than $6.5$
For the LHS:
Union of two sets has the elements of both the sets.
So, $B\cup C=$ { $1,2,3,4,5$ }
Intersection of two sets has the elements common in both the sets.
$=>A\cap (B\cup C)=$ { $1,2,3,4,5$ } --- (1)
For the RHS:
$A\cap B=$ { $3$ }
$A\cap C=$ { $1,2,3,4,5$ }
$=>(A\cap B)\cup (A\cap C)=$ { $1,2,3,4,5$ } --- (2)
From, 1 and 2
$=>A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$