Write the negation of the following statements.(i) A triangle is equil

1.	Write the negation of the following statements.(i) A triangle is equilateral if and only if it is equiangular.(ii) Sets A and B are equal if and only if A ≤ B and B ≤ A.
Answer» (i) A triangle is equilateral if and only if it is equiangular. Let, p ∶ a triangle is equilateral. q : A triangle is equiangular. ~ (p ⇒ q) ≡ (p ∩ ~q) ∪ (q ∩ ~ p) The negation of the statement is “There exists either an equilateral triangle which is not equiangular or on equiangular triangle which is not equilateral. (ii) Sets A and B are equal if and only if A ≤ B and B ≤ A. Let, p : Sets A and B are equal. q ∶ A ≤ B and B ≤ A. ~ (p(=)q) ≡ (p ∩ ~q) ∪ (q ∩ ~ p) The negation of the statement is- Either A = B and (A ≤ B or B ≤ A.) or (A ≤ B and B ≤ A.) and ≠ B.”

Write the negation of the following statements.(i) A triangle is equilateral if and only if it is equiangular.(ii) Sets A and B are equal if and only if A ≤ B and B ≤ A.

Answer»

(i) A triangle is equilateral if and only if it is equiangular.

Let, p ∶ a triangle is equilateral.

q : A triangle is equiangular.

~ (p ⇒ q) ≡ (p ∩ ~q) ∪ (q ∩ ~ p)

The negation of the statement is

“There exists either an equilateral triangle which is not equiangular or on equiangular triangle which is not equilateral.

(ii) Sets A and B are equal if and only if A ≤ B and B ≤ A.

Let, p : Sets A and B are equal.

q ∶ A ≤ B and B ≤ A.

~ (p(=)q) ≡ (p ∩ ~q) ∪ (q ∩ ~ p)

The negation of the statement is-

Either A = B and (A ≤ B or B ≤ A.) or (A ≤ B and B ≤ A.) and ≠ B.”