Question

AX (B intersection C) = (AXB) intersection (AXC) prove it​

Answer 1

GIVEN :

$A\times (B\bigcap C)=(A\times B)\bigcap (A\times C)$ prove it.

TO PROVE :

The given equality $A\times (B\bigcap C)=(A\times B)\bigcap (A\times C)$ is true

SOLUTION :

Given equation is $A\times (B\bigcap C)=(A\times B)\bigcap (A\times C)$

We have to prove that the given equation is true.

ie., LHS = RHS :

We know that the definition of $A\times B$ as  below :

$A\times B={\{(x,y)|x\epsilon A,y\epsilon B}\}$

Now taking the LHS,

$A\times (B\bigcap C)$

$={\{(x,y)|x\epsilon A and y\epsilon B\bigcap C}\}$

$={\{(x,y)|x\epsilon A and y\epsilon B and y\epsilon C}\}$

$={\{(x,y)|(x,y)\epsilon A\times B and (x,y)\epsilon A\times C}\}$

$=(A\times B)\bigcap (A\times C)$

= RHS

⇒ LHS = RHS

∴ $A\times (B\bigcap C)=(A\times B)\bigcap (A\times C)$

∴ the given equation $A\times (B\bigcap C)=(A\times B)\bigcap (A\times C)$ is true and is verified.