Prove that A and B are disjoint

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In summary, to prove that two sets A and B are disjoint, you can either show that if an element is in A, then it is not in B, or if an element is in B, then it is not in A. It is not necessary to prove both directions. Munkres (Topology) offers a proof for this concept in Chapter 2 or 3.
  • #1
alexmahone
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How do I prove that 2 sets A and B are disjoint?

Do I let $\displaystyle x\in A$ and prove that $\displaystyle x\notin B$? Do I also have to prove the converse?
 
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  • #2
Alexmahone said:
How do I prove that 2 sets A and B are disjoint?
Do I let $\displaystyle x\in A$ and prove that $\displaystyle x\notin B$? Do I also have to prove the converse?
As stated, the question is too vague to give a clear answer.
That said, you might suppose that [tex]\exists x\in A\cap B[/tex].
Prove that leads to a contradiction.
 
  • #3
Plato said:
As stated, the question is too vague to give a clear answer.
That said, you might suppose that [tex]\exists x\in A\cap B[/tex].
Prove that leads to a contradiction.

Thanks for your answer, but why is the question too vague?
 
  • #4
Alexmahone said:
Thanks for your answer, but why is the question too vague?
Because you told us nothing about $A\text{ or }B$.
 
  • #5
Plato said:
Because you told us nothing about $A\text{ or }B$.

A and B are 2 arbitrary sets.
 
  • #6
Alexmahone said:
A and B are 2 arbitrary sets.
[tex]\left( {\forall x \in \mathcal{ U}} \right)\left[ x \notin A\cap B \right][/tex].
 
  • #7
if for ALL x in A, x is not in B, that would be sufficient. why? because that asserts that A is a subset of A - B = A - A∩B. since A - B is automatically a subset of A, we get immediately that A = A - B = A - A∩B, hence A∩B = Ø (if y was in A∩B, then y would not be in A-B, a contradiction since A = A-B).

however, you have to be careful. just picking "some" x in A, and showing it is not in B, just shows x is in A-B, which can happen when A and B are not disjoint. the choice of x has to be completely arbitrary (a "for all" choice, not a "there exists" choice).

for example, let's use "your method" to prove (0,1) and (3,4) (these are real intervals) are disjoint. let x be any real number in (0,1). then 0 < x < 1. since 1 < 3, by the transitivity of < we have x < 3. hence (x > 3)&(x < 4) is false (trichotomy property: exactly one of x < 3, x = 3 or x > 3 can be true...note this is a specific property of real numbers, we are using the fact that R is an ordered field, here), that is: 3 < x < 4 is false, so x is not in (3,4). since x is arbitrary, we conclude (0,1) and (3,4) are disjoint.

the following is a "bad proof": let x be in (0,1). then x is not in (3,4), so (0,1) and (3,4) are disjoint. why is this bad? because x might be 1/2, and all we have shown is 1/2 is not in (3,4).

it's a subtle difference, and Plato's posts are meant to underscore the important part: disjoint sets don't "overlap". put another way: quantification matters (logically speaking).
 
  • #8
It is sufficient to prove "if x is in A then it is NOT in B" or "if x is in B then it is NOT in A". You do not have to prove both.
 
  • #9
Alexmahone said:
How do I prove that 2 sets A and B are disjoint?

Do I let $\displaystyle x\in A$ and prove that $\displaystyle x\notin B$? Do I also have to prove the converse?

In Munkres (Topology), there is a proof about showing sets are disjoint which is probably along the lines you are looking for. However, I can't remember off hand but it may not be too set theoretic. Chapter 2 or 3 I believe. It will be on a left hand side page at the bottom continuing to right page on the top.
 

FAQ: Prove that A and B are disjoint

What does it mean for two sets to be disjoint?

Two sets A and B are disjoint if they have no common elements. In other words, the intersection of A and B is equal to the empty set.

How can you prove that two sets are disjoint?

To prove that two sets A and B are disjoint, you can show that their intersection is equal to the empty set. This can be done by checking if there are any elements that are present in both sets. If there are no common elements, the sets are disjoint.

Can two sets with the same elements be disjoint?

No, two sets with the same elements cannot be disjoint. In order for two sets to be disjoint, they must have no common elements. If two sets have the same elements, then they have at least one common element, making them not disjoint.

Is it possible for three or more sets to be disjoint?

Yes, it is possible for three or more sets to be disjoint. As long as all the sets have no common elements, they can be considered disjoint.

How do you use Venn diagrams to represent disjoint sets?

Venn diagrams can be used to visually represent disjoint sets by drawing two or more circles that do not overlap. Each circle represents a set and the area where the circles do not overlap represents the empty set, indicating that the sets are disjoint.

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