# How do I prove this? Sets

1. Oct 4, 2011

### flyingpig

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

For the sets A and B, prove that

$$A \cap B \subseteq A \subseteq A \cup B$$

3. The attempt at a solution

I am guessing I should look at only two of them first?

$$A \subseteq A \cup B$$

What conditions do I need?

2. Oct 4, 2011

### SammyS

Staff Emeritus
Definition of C ⊆ D ?

Let x ∊ A⋂B, then ...

3. Oct 4, 2011

### Punkyc7

let x be in A intersect B..what does that mean...

4. Oct 4, 2011

### flyingpig

C is a subset of D

if x is in the intersection of A and B, then x belongs to both A and B

5. Oct 4, 2011

### Punkyc7

so x is in A .... is x in the Union of A and B

6. Oct 4, 2011

### SammyS

Staff Emeritus
∴ x belongs to A .

7. Oct 4, 2011

### flyingpig

Ohhhh

so for

$$A \subseteq A \cup B$$

Same argument? i.e.

$$A \cup B$$ for some element x, belongs to A or B and hence A also belongs to A? DOes the "or" say whether it can have elements in A or not? Is it a hasty conclusion?

8. Oct 4, 2011

### Punkyc7

yeah your elements could be in either A or B, it helps to draw a picture

9. Oct 4, 2011

### SammyS

Staff Emeritus
If x is in A, then x is in (A or B).

10. Oct 5, 2011

### flyingpig

I want to do this elegantly

So

11. Oct 5, 2011

### flyingpig

I just want to ask, I don't need to show that $$A \cap B \subseteq A \cup B$$ right? Because this just follows from subset properties? Does this make a good proof?