Write the negation of the following. If x ∈ A ∩ B, then x ∈ A and x ∈ B.

QUESTION

Miscellaneous Exercise 1 | Q 4.1 | Page 32

Write the negation of the following.

If x ∈ A ∩ B, then x ∈ A and x ∈ B.

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.