Let A ,B and X be three sets such that `A cap X= B cap X = phi and A cup X = B cup X` .Then show that A=B .

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.