1） Use the logical equivalences below and the definitions of set operations to prove, or provide a counterexample to disprove, the following set identity: (A ∪ B) ∖ C = (A ∖ C) ∪ (B ∖ C) 2）Use the logical equivalences below and the definitions of set operations to prove, or provide a counterexample to disprove, the following set identity: (A ∪ B) ∖ C = (A ∖ C) ∪ B
Question
1） Use the logical equivalences below and the definitions of set operations to prove, or provide a counterexample to disprove, the following set identity: (A ∪ B) ∖ C = (A ∖ C) ∪ (B ∖ C) 2）Use the logical equivalences below and the definitions of set operations to prove, or provide a counterexample to disprove, the following set identity: (A ∪ B) ∖ C = (A ∖ C) ∪ B
Solution
- To prove the set identity (A ∪ B) ∖ C = (A ∖ C) ∪ (B ∖ C):
The left-hand side (A ∪ B) ∖ C represents the set of all elements that are in A or B but not in C.
The right-hand side (A ∖ C) ∪ (B ∖ C) represents the set of all elements that are in A but not in C, or in B but not in C.
These two sets are equivalent because in both cases, we are including all elements that are in A or B but excluding those that are in C. Therefore, (A ∪ B) ∖ C = (A ∖ C) ∪ (B ∖ C).
- To disprove the set identity (A ∪ B) ∖ C = (A ∖ C) ∪ B:
Consider the following counterexample:
Let A = {1, 2}, B = {2, 3}, and C = {2}.
Then, (A ∪ B) ∖ C = ({1, 2, 3}) ∖ {2} = {1, 3}.
However, (A ∖ C) ∪ B = ({1, 2} ∖ {2}) ∪ {2, 3} = {1} ∪ {2, 3} = {1, 2, 3}.
Clearly, {1, 3} ≠ {1, 2, 3}. Therefore, the set identity (A ∪ B) ∖ C = (A ∖ C) ∪ B is not always true.
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