Question

Let A and B be sets. If $A ∩ X = B ∩ X = \phi$ and $A ∪ X = B ∪ X$ for some set X, show that A = B. (Hints $A = A ∩ (A ∪ X), B = B ∩ (B ∪ X)$ and use distributive law)

Answer 1

Let A and B be two sets such that $A ∩ X = B ∩ X = f$ and $A ∪ X = B ∪ X$ for some set X.
To show: $A = B$
It can be seen that

$A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X]$
$= (A ∩ B) ∪ (A ∩ X)$ $[$Distributive law] = $(A ∩ B) ∪ \phi [A ∩ X = \phi]$
$= A ∩ B$ …………………………………………………………….. (1)

Now, $B = B ∩ (B ∪ X)$
$= B ∩ (A ∪ X) [A ∪ X = B ∪ X]$
$= (B ∩ A) ∪ (B ∩ X)$ [Distributive law]
$= (B ∩ A) ∪ \phi [B ∩ X = \phi]$
$= B ∩ A$
$= A ∩ B$ …………………………………………………………… (2)

Hence, from (1) and (2), we obtain A = B.