#1: Prove or provide a counterexample:
For all sets A, B, C, if A is subset of B and B is a subset of C^c (complement of C), then AC= { }.
This is my proof so far, but I don't know if it is correct?
Assume to the contrary that A ⋂ C = { }. This means that there exists x∈(A⋂C). This then means that x∈A and x∈C. By our starting assumption, if x∈A, then x∈B because A⊆B. Also, if x∈B, then x∈C^C given that B⊆C^C. This shows that there is a contradiction because x∈C and x∈C^C. That means that A⋂C = { } is false. Therefore,the original statement of A⋂C = { } is true.
Your proof is almost correct, but there are a few issues that need to be addressed. Here is a step-by-step explanation of how to approach this proof correctly:
1. Start by assuming that A ⊆ B and B ⊆ C^c (complement of C).
2. Now, assume to the contrary that AC is not equal to the empty set, i.e., AC ≠ {}. This means that there exists an element x ∈ AC.
3. By the definition of set difference, x ∈ AC means that x ∈ A and x ∉ C.
4. Since x ∈ A and A ⊆ B, we can conclude that x ∈ B.
5. Also, since x ∉ C and B ⊆ C^c, we can conclude that x ∈ C^c.
6. Recall that the complement of C, C^c, is defined as the set of elements that are not in C. Therefore, if x ∈ C^c, then x ∉ C.
7. Combining the last two statements, we have reached a contradiction: x ∈ C and x ∉ C. This is not possible.
8. Hence, our assumption that AC ≠ {} is incorrect, and we can conclude that AC = {}.
Therefore, the statement "if A is a subset of B and B is a subset of C^c, then AC = {}" is true.
It's important to note that in this proof, we assumed that A ∩ C ≠ {}. Your original proof attempted to prove this assumption false, but it was not necessary. The contradiction could be found even without assuming A ∩ C ≠ {}.