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.
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.
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 ...you never stated what you are assuming for you to get to thisThen There exists (x that belongs to C∩D) and (x does not belong to A∩ B)
okSo x belongs to C and x belongs to D
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 theseIf 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.
sorry, but this requires a complete do over.
start by saying what you are assuming. begin, for example, as follows:
Proof:
Assume the . then there is some element that is not in . But that means ...