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?