Question

If A={1,2,3}, B={4}, C={5} then verify that:
(1). Ax (B UC) = (AxB)U(AXC) (2). A x(B-C) = (AxB)-(AXC)

Answer 1

GIVEN :

The sets are  A={1,2,3}, B={4}, C={5}

TO VERIFY :

(1). $A\times (B\cup C) = (A\times B)\cup (A\times C)$

(2). $A\times (B-C) = (A\times B)-(A\times C)$

SOLUTION :

Given sets are A={1,2,3}, B={4}, C={5}

Now verify that

(1). $A\times (B\cup C) = (A\times B)\cup (A\times C)$

Taking LHS  $A\times (B\cup C)$

$B\cup C={\{4,5}\}$

$A\times (B\cup C)={\{1,2,3}\}\times {\{4,5}\}$

$A\times (B\cup C)= {\{ ( 1 , 4 ) , ( 1 , 5 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 3 , 4 ) , ( 3 , 5 ) }\}$= LHS

Taking RHS $(A\times B)\cup (A\times C)$

$A\times B= {\{ ( 1 , 4 ) , ( 2 , 4 ) , ( 3 , 4 )}\}$

$A\times C= {\{ ( 1 , 5 ) , ( 2 , 5 ) , ( 3 , 5 )}\}$

$(A\times B)\cup (A\times C) {\{ ( 1 , 4 ) , ( 1 , 5 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 3 , 4 ) , ( 3 , 5 ) }\}$= RHS

∴ LHS = RHS

∴ $A\times (B\cup C) = (A\times B)\cup (A\times C)$ is verified.

(2). $A\times (B-C) = (A\times B)-(A\times C)$

Taking LHS  $A\times (B-C)$

$B - C={\{4}\}$

$A\times (B-C)={\{( 1 , 4 ) , ( 2, 4) , (3, 4)}\}$=LHS

Taking RHS $(A\times B)-(A\times C)$

We know $A\times B= {\{ ( 1 , 4 ) , ( 2 , 4 ) , ( 3 , 4 )}\}$

$A\times C= {\{ ( 1 , 5 ) , ( 2 , 5 ) , ( 3 , 5 )}\}$

$(A\times B)-(A\times C)={\{ ( 1 , 4 ) , ( 2 , 4 ) , ( 3 , 4 )}\}- {\{ ( 1 , 5 ) , ( 2 , 5 ) , ( 3 , 5 )}\}$

$(A\times B)-(A\times C)={\{ ( 1 , 4 ) , ( 2 , 4 ) , ( 3 , 4 )}\}$=RHS

∴ LHS = RHS

∴ $A\times (B-C) = (A\times B)-(A\times C)$ is verified.