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

1.	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» 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.