Proving Set Inclusion: A \subseteq A \cup B

[flyingpig]

Homework Statement

For the sets A and B, prove that

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

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?

 

[SammyS]

Definition of C ⊆ D ?

Let x ∊ A⋂B, then ...

 

[Punkyc7]

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

 

[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

 

[Punkyc7]

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

 

[SammyS]

C is a subset of D

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

∴ x belongs to A .

 

[flyingpig]

∴ x belongs to A .

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?

 

[Punkyc7]

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

 

[SammyS]

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

 

[flyingpig]

I want to do this elegantly

So

$$x \in A \cap B \iff x \in A$$ and $$x \in B$$

$$\therefore x \in A$$

So $$A \cap B \subseteq A$$

$$x \in A \cup B \iff x \in A$$ or $$x \in B$$

So $$A \subseteq A \cup B$$

q.e.d

 

[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?