Proving A=B if A & B are Subsets

[cragar]

Homework Statement

Prove that if A=B if and only if $A \subseteq B$ and $B \subseteq A$

The Attempt at a Solution

If A is a subset of B then all the elements of A are in B . And if B is a subset of A then all the elements of B are in A . There fore there is a one-to-one correspondence between the 2 sets therefore they are equal. Is this to simple or do I need to be more rigorous.

 

[gb7nash]

You might want to have a little more rigor. Do an element-wise proof.

e.g.

(<-) Fix x in A. Since A is a subset of B, ...

Similarly, ...

So, ...

This only proves the backwards direction.

 

[Deveno]

you aren't looking for ANY old 1-1 correspondence, you're looking for EQUALITY.

if x is in A, and A ⊆ B, then x is in B.

so B has every element A does (maybe more).

but since B ⊆ A, there cannot be any element of B that is not in A

(if there was, we quickly get a contradiction).

so B has exactly the same elements as A, thus A = B.

now, if A = B, why is A ⊆ B true?

 

[cragar]

A subset can be equal to the set itself .

 

[vela]

What's the definition of set equality?

 

[cragar]

they have the same numbers of elements or the same cardinality.

 

[vela]

So {1,2,3} and {3,4,5} are equal?

 

[cragar]

ok so they have to have the same elements

 

[vela]

So what specifically do you have to show to prove that two sets have the same elements?

 

[cragar]

thanks for you help by the way . I have to show everything in A is in B and everything in B is in A . Ill work on it .

 

[vela]

You just keep saying the same thing with different words, but there's a precise way to say "everything in A is in B" and vice versa. That's one thing you'll need to know to write a proper proof. So start by looking up what the precise definition of set equality is.

 

[HallsofIvy]

thanks for you help by the way . I have to show everything in A is in B

Which is exactly the same thing as saying $A\subseteq B$.

and everything in B is in A

Which is exactly the same thing as saying $B\subseteq A$.

. Ill work on it .

 

[cragar]

if $x \in A$ then $x \in B$
if $x \in B$ then $x \in A$
therefore A=B
It seems so simple. But I am not sure if that's good enough.
Or am I going in circles .

 

[vela]

It's not complete. When you have an if and only if, you have to prove both directions. In this case, you need to prove:

(1) If A⊂B and B⊂A, then A=B.
(2) If A=B, then A⊂B and B⊂A.

You've essentially shown (1). Part of your confusion probably comes from thinking you're not really showing anything; that is, it's too obvious. With these really elementary proofs, you want the mindset that you can take nothing for granted and need to justify every little step, no matter how insignificant it may seem. If I were you, I'd add the step in red below to explicitly explain how you can conclude A=B.

Assume A⊂B and B⊂A. By definition,

A⊂B means (x∈A → x∈B)
B⊂A means (x∈B → x∈A)

Hence, we have x∈A ⟺ x∈B. Therefore, by definition of set equality, we can conclude A=B.

[I'm assuming your definition of set equality is: A=B iff (x∈A ⟺ x∈B).]

 

[cragar]

ok I see , Thanks you for our help.