Proving Element In Union of Two Infinite Sets Not Necessarily In Intersection

[woundedtiger4]

Problem:
Prove that if an element is in the union of two infinite sets then it is not necessarily in their intersection:

Proof:

Have I solved it correctly?

 

[micromass]

Looks good.

 

[HallsofIvy]

Correct but awkward. The best way to prove something is NOT true is to give a counter-example. Let A= {(x, 0)}, the set of all points on the x-axis. Let B= {(0, y)}, the set of all points on the y-axis. Those are both infinite sets and their union is the set of all pairs of numbers in which at least one of the pair is 0. But their intersection is just {(0, 0)}.

 

[skiller]

Correct but awkward.

I don't see the OP's method being any more or less awkward than yours, really.

The best way to prove something is NOT true is to give a counter-example.

That is exactly what the OP did; just using a different couple of sets. The main difference was that the OP's two sets didn't even intersect at all, which is fine.

In fact, you could argue that that is a more obvious way to find a counter-example. Surely it is easier to find two infinite sets that do not intersect in the first place?

However, I'm not saying your answer was awkward either.

 

[blue_raver22]

Yes, your proof is correct. To prove that an element is not necessarily in the intersection of two infinite sets, we can use a counterexample. Let's consider the sets A = {1, 2, 3, ...} and B = {2, 4, 6, ...}. Both of these sets are infinite and their union is also infinite. However, their intersection is the empty set, as there are no common elements between them. Therefore, any element that is in the union of A and B (e.g. 2) is not necessarily in their intersection. This counterexample proves that the statement "if an element is in the union of two infinite sets then it is not necessarily in their intersection" is true.