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Question
Write the negation of the following.
If x ∈ A ∩ B, then x ∈ A and x ∈ B.
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Solution
Let p : x ∈ A ∩ B
q : x ∈ A
r : x ∈ B
The given statement is p → (q ∧ r).
Its negation is ~[p → (q ∧ r)], and
~[p → (q ∧ r)] ≡ p ∧ ~ (q ∧ r) ≡ p ∧ ~ q ∨ ~ r
∴ The negation of given statement is x ∈ A ∩ B and x ∉ A or x ∉ B.
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