Let A and B be sets. If A`cap`X=B`cap`X=`phi` and A`cup`X=B`cup`X for some set X, show that A=B.
(Hints A=A`cap`(A`cup`X), B=B`cap`(B`cup`X) and use distributive law)

Explanation

We have `A cap B = B cap X = phi " and " A cup X = B cup X`
It can be seen that
`A= A cap ( A cup X) =A cap ( B cup X)" " [ A cup X B cup X ] `
`= (A cap) cup (A capX) " " `[Distributive law]
`= (A cup B) cap phi " " [A cup X = phi]`
`= A cup B " " ... (1)`
Now , `B = B cup (B cap X)`
`= B cup (A cap X) " " [ A cap X = B cap X]`
= `(B cup A ) cap (B cup X) " " ` [Distributive law]
`= (B cup A) cap phi " " [B cup X = phi]`
`= B cup A `
= `A cup B " " .. (2)`
Hence from (1) and (2) , we obtain A = B.