Prove Intersection of A and B is Empty Set if B Subset of X/A

• chocolatelover
In summary, to prove that A intersects B = empty set if and only if B is a subset of (X/A), one must prove the statement both ways using either a proof by contradiction or selecting elements of certain sets and logically showing their existence or non-existence in other sets.
chocolatelover

Homework Statement

Prove A intersects B=empty set if and only if B is a subset of (X/A)

The Attempt at a Solution

Would I prove the contrapositive in this case?

If B is not a subset of (X/A), then the intersection of A at B is not the empty set

Could someone please show me what to do?

Thank you very much

"if and only if" means that you must prove the statement both ways. Proving by contradiction seems the easiest way for each. The approach you suggest will only get you half the proof.

Thank you very much

Could you show me where to go from there?

Thank you

Well since you wanted to use the contrapositive for A intersects B = empty -> B is a subset of X\A I'll explain that way. Let $B=\{b_1,b_2,...\}$. If B is empty what we want is vacuously true since the empty set is a subset of every set (and thus B can never not be a subset of $X \setminus A$). If $B\not\subset X\setminus A$ then at least one $b_i \in B$ is $\in X\setminus (X \setminus A) = A$ which implies A and B have these elements in common.

You can prove the other way in a similar fashion. My personal suggestion is contradiction for the other way.

On this note, you have another question that's pretty similar. Your questions really boil down to choosing elements of certain sets and then showing by logic that they must/must not exist in other sets. Try to proceed like this in your other question also.

Last edited:
Thank you very much

Regards

What does it mean for B to be a subset of X/A?

When we say that B is a subset of X/A, it means that every element in B is also in X/A. In other words, B is a smaller set that is completely contained within X/A.

What is the intersection of A and B?

The intersection of A and B is the set of all elements that are common to both A and B. In other words, it is the set of elements that are in both A and B.

Why is it important to prove that the intersection of A and B is an empty set?

This is important because it helps to establish that A and B have no common elements. This can be useful in various mathematical and scientific contexts, such as when trying to prove the uniqueness of a solution or when studying the relationships between different sets.

What does it mean for the intersection of A and B to be an empty set?

An empty set is a set that contains no elements. So, if the intersection of A and B is an empty set, it means that there are no elements that are common to both sets A and B.

How do you prove that the intersection of A and B is an empty set if B is a subset of X/A?

First, we need to understand that if B is a subset of X/A, then all elements in B are also in X/A. Therefore, the only way for the intersection of A and B to be empty is if there are no elements in A that are also in B. This can be proven by assuming that there is an element in A that is also in B and showing that this leads to a contradiction. Thus, we can conclude that the intersection of A and B must be an empty set.

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