# If and only if Proof

1. Sep 9, 2007

### mutzy188

1. The problem statement, all variables and given/known data

Prove or give a counterexample to the statement:

S ∪ T = T ↔ S ⊆ T

3. The attempt at a solution

What I did:

Let S={1,2,3,4} and T = {1,2}

S ∪ T = {1,2} = T

S ⊆ T

{1,2,3,4} ⊈ {1,2}

Therfore it is False . . .but the answer in the book says that it is true

Thanks

2. Sep 9, 2007

### D H

Staff Emeritus
You are confusing union and intersection. The intersection of {1,2,3,4} and {1,2} is {1,2} but their union is {1,2,3,4}.

3. Sep 9, 2007

### dextercioby

HINT: $S\cup T\subset T$.

4. Sep 9, 2007

### d_leet

This isn't tru in general for any S and T, for example let T={1,2,3,4}, and S={1,5} then SUT={1,2,3,4,5} which is not a subset of T. It is true, however, if you replace the union with intersection.

EDIT: It's also true if you change the direction of inclusion to say that T is a subset of SUT.

5. Sep 10, 2007

### AsianSensationK

You missed the point of the hint. It's true in this problem because you're given that S U T = T. It follows from the definition of equality.

He gave you the first step to the proof. Now you have to ask what that says about the relationship between S and T?