How Do You Prove Set Relations Involving Subsets and Unions?

[arpitm08]

Proof involving sets. NEED HELP!

Homework Statement

Prove directly "If A U B = A, then B is a subset of A." and also provide a proof by contrapositive of its converse.


2. The attempt at a solution

Here is what i did, but I don't know if it is right or not,

Direct Proof: Assume A U B = A, then x ∈ (A U B) and x ∈ A. So it follows that B ∈ A = B is a subset of A.
Contrapositive of Converse Proof: Assume that A U B ≠ A, then x ∈ (A U B) and x ∉ A. Then, B ∉ A and so B is not a subset of A.

I don't think this is right. Could someone help me out please??

 

[J$C]

Direct Proof: Assume A U B=A as you have. Then to show a set B is a subset of a set A the standard technique is to let x be in B, then show it is also in A. Notice if x is in B then it is clearly also in A U B and the conclusion follows from your initial assumption.

Converse: If B is a subset of A then A U B=A
Contrapositive of Converse: If A U B /neq A then B is not a subset of A

Notice A U B /neq A but A is clearly a subset of A U B. So what's left to make that not equals is A U B is not a subset of A. That gives that there's an element in A U B that is not in A. Go from there.

 

[arpitm08]

How about now...

Directly -
Assume A U B = A, then x ∈ B. Then x ∈ (A U B), and since A U B = A, x ∈ A. So B is a subset of A.

Contrapositive of Converse -
Assume that A U B ≠ A. Since A is a subset of A U B, there must be an x ∈ (A U B), such that x ∉ A, since A U B ≠ A. This means that there is a y ∈ B, such that y ∉ A. So B is not a subset of A.

Is that a complete proof??

 

[lanedance]

in contrapositive of converse

though its implied, I think you should change it to A is a proper subset of AUB

 

[arpitm08]

Allright. Thanks! =D

 

[J$C]

Looks good.