Proof Help: Prove (A∪B)-C ⊆ A∪(B-C)

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In summary, The conversation is about a proof question where the goal is to prove that one set is a subset of another. The suggested approach is to take an arbitrary element from the first set and prove that it also belongs to the second set. The person asking for help is preparing for an exam the next day and is grateful for the assistance in understanding the proof.
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Ja6464
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I am stuck on a proof question:

Prove (A∪B)-C ⊆ A∪(B-C)

If anyone would be able to help me with this proof it would be much appreciated, as I have an exam on this topic tomorrow afternoon!

Thanks a lot.
 
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  • #2
In a proof like this (checking if a set is included into another) you typically take one arbitrary element x from the first set, and prove that it also belongs to the second set. (So any element from the first is included in the second, so the first set is a subset of the second.)

If x is in the first set, then it is either in A or in B (or both), but certainly not in C. Try to figure out each of the two cases (x in A, x in B), and see if, in both cases, x belongs to the second set.

Hope this helps.
 
  • #3
Thank you that's helped a lot, I think I've managed to prove it now.

Hopefully I can do it again tomorrow!
 

1. What is the meaning of "Proof Help: Prove (A∪B)-C ⊆ A∪(B-C)"?

The statement (A∪B)-C ⊆ A∪(B-C) means that the elements in the set (A∪B) that are not in the set C are also contained in the set A∪(B-C).

2. How do you read the "∪" symbol in the statement?

The "∪" symbol represents the union of two sets, meaning all the elements in both sets are combined into one set.

3. What is the purpose of proving this statement?

The purpose of proving this statement is to show that the set (A∪B)-C is a subset of the set A∪(B-C). This can help in solving more complex problems in mathematics and other fields that involve set theory.

4. What is the first step in proving this statement?

The first step in proving this statement is to assume an arbitrary element x in the set (A∪B)-C and then show that x is also in the set A∪(B-C).

5. Can you provide an example to illustrate this statement?

Let A = {1, 2, 3}, B = {3, 4, 5}, and C = {1, 5}. The set (A∪B)-C would be {2, 3, 4}, and the set A∪(B-C) would be {1, 2, 3, 4}. The statement (A∪B)-C ⊆ A∪(B-C) is proven by showing that all the elements in (A∪B)-C (2, 3, and 4) are also in the set A∪(B-C).

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