Question

Prove that A∩(AUB)=A
Do I have have to show that A∩(AUB)⊂A and that A⊂A∩(AUB)????

Answer 1

if \(A\subset C\) then \(A\cap C=A\)

Answer 2

so I do.

Answer 3

use venn diagram..much easier to prove sets identities

Answer 4

it is sufficient to show that A ⊂(A∩(AUB)) and A∩(AUB) is a subset of A.
first we know that AUB will have all the elements of A, so that intersection of A and AUB will also contain all the elements in A, hence A is a subset of A∩(AUB). Now since AUB and intersection of it with A will only have elements of A (because if it contains elements which are not in A, then it will not be intersection of A and AUB lol.) so A∩(AUB) is a subset of A.
hence A∩(AUB)=A