Is A Contained in B Equivalent to A Intersecting B Equals A?

[mbcsantin]

Homework Statement

Prove A contained B iff A n B = A

Homework Equations

none

The Attempt at a Solution

I tried to do the questions but I am just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks

the symbol "n" means "intersect"
U for Union


(=>) Assume A contained B

Let x is an element of A, since A n A = A, x is an element of A and x is an element of B

Case 1: x is an element of A: Since A contained B, x is an element of B so
x is an element of A n B

Case 2: x is an element of B: If x is an element of B then
x is an element of (A n B)

Hence x is an element of A n B

This shows A contained A n B

(<=) Assume A n B = A then

A’=A’UA
= A’ U (A n B)
=(A’UA) n (A’U B)
= empty set n A’ U B
= A’ U B

Hence
Universe = A’ U B

 

[e(ho0n3]

Let x is an element of A, since A n A = A, x is an element of A and x is an element of B

You don't need the "since A n A = A" part.

Case 1: x is an element of A: Since A contained B, x is an element of B so
x is an element of A n B

Case 2: x is an element of B: If x is an element of B then
x is an element of (A n B)

Hence x is an element of A n B

This shows A contained A n B

You don't need cases here. You want to show that A n B = A: Do this by first showing that A n B is a subset of A and then showing that A is a subset of A n B. (Do you see why this implies A n B = A?)

(<=) Assume A n B = A then

A’=A’UA
= A’ U (A n B)
=(A’UA) n (A’U B)
= empty set n A’ U B
= A’ U B

Hence
Universe = A’ U B

What is A' exactly? Also, you state that A’=A’UA but then you have that A’=A’UA is the empty set. Surely there is something wrong here. You just need to show that A is a subset of B. Do this by picking a random member of A and show that it also belongs to B.