Prove that
A×(BUC)=(A×B)U(A×C)
Answers
Answer:
A×(B U C) =(A×B)U (A×C)
Step-by-step explanation:
To prove ---->
---------------
A×(B U C) =(A×B) U (A×C)
concept --->
--------------
if (x, y) € P×Q
=> x€P , y€ Q
if x € P U Q
means
x€P or X € Q
now returning to original problem
proof--->
-----------
let
(x, y) € A×(B U C)
=> x€ A and y€(B U C)
=> x €A and (y€B or y€C)
=>(x€A and y€B) or(x€A and y€C)
=>(x, y) €(A×B) or (x, y) €(A×C)
=>(x, y) € {(A×B) U (A×C)}
so it means
A×(B U C) is subset or equal set to
(A×B) U (A×C) ---------------(1)
now let
(a, b)€ (A×B) U (A×C)
=> (a,b)€(A×B) or (a, b) €(A×C)
=>(a€A and b€B )or (a€A and b€C)
=>a€A and (b€B or b€C)
=>a€A and b€ (B U C)
=>(a,b) € A×(B U C)
it means
(A×B) U(A×C) is subset or equal set to
A×(B U C) -----------------(2)
by (1) and (2) we get
A×(B U C) = (A×B) U (A×C)
hence proved
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