Thread: Set theory: One more quick proof (low level)

1. Set theory: One more quick proof (low level)

A,B,C,D are sets.

Prove that if C is contained in A and D is contained in B, then C∩ D is contained in A∩ B.

Would proof by contrapositive be appropriate?

I.e. show that if the first intersection is not contained in the second, then C not contained in A and so on or is a different method better? I just have trouble formally showing these things.

2. Originally Posted by zhupolongjoe
A,B,C,D are sets.

Prove that if C is contained in A and D is contained in B, then C∩ D is contained in A∩ B.

Would proof by contrapositive be appropriate?

I.e. show that if the first intersection is not contained in the second, then C not contained in A and so on or is a different method better? I just have trouble formally showing these things.
That sounds like an excellent idea! If C∩D is NOT contained in A∩B, there exist x in C∩D that is not in A∩D and there for x is in C and x is in D. Be sure you don't say "because x is not in A∩D, it is in neither A nor D"- that is not true.

3. Ok thanks, is this an acceptable proof:

Let x be any element.

Then There exists (x that belongs to C∩D) and (x does not belong to A∩ B)

So x belongs to C and x belongs to D

If x belongs to C, since C is contained in A, then x belongs to A.
If x belongs to D, since D is contained in B, then x belongs to B.
So x belongs to A intersect B, a contradiction.

Then the original statement is true.

4. Originally Posted by zhupolongjoe
Ok thanks, is this an acceptable proof:

Let x be any element.
any element? no, that won't work. that statement is too vague to be of any use. you mean let x be any element of ...
Then There exists (x that belongs to C∩D) and (x does not belong to A∩ B)
you never stated what you are assuming for you to get to this

So x belongs to C and x belongs to D
ok

If x belongs to C, since C is contained in A, then x belongs to A.
If x belongs to D, since D is contained in B, then x belongs to B.
So x belongs to A intersect B, a contradiction.
when were we given that C is contained in A and D is contained in B? we are using the contrapositive right? that means, we cannot use these

sorry, but this requires a complete do over.

start by saying what you are assuming. begin, for example, as follows:

Proof:
Assume the $\displaystyle C \cap D \not \subseteq A \cap B$. then there is some element $\displaystyle x \in C \cap D$ that is not in $\displaystyle A \cap B$. But that means ...