The negation of  is equivalent to

The statement ~ (p↔ ~q) is 

• equivalent to p ↔ q

• equivalent to ~ p ↔q

• a tautology

• a tautology

The variance of first 50 even natural number is

• 833/4

• 833

• 437

• 437

Consider :
Statement − I : (p ∧ ~ q) ∧ (~ p ∧ q) is a fallacy.
Statement − II : (p → q) ↔ (~ q → ~ p) is a tautology.

• Statement -I is True; Statement -II is True; Statement-II is a correct explanation for Statement-I 

• Statement - I is True; Statement -II is true; Statement-II is not a correct explanation for Statement-I

• Statement -I is True; Statement -II is False.

• Statement -I is True; Statement -II is False.

The negation of the statement “If I become a teacher, then I will open a school” is

• I will become a teacher and I will not open a school

• Either I will not become a teacher or I will not open a school

• Neither I will become a teacher nor I will open a school

• Neither I will become a teacher nor I will open a school

Consider the following statements
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest. The negation of the statement ì Suman is brilliant and dishonest if and only if Suman is richî can be ex- pressed as

•  ~ P ^ (Q ↔ ~ R)

• ~ (Q ↔ (P ^ ~R)

• ~ Q ↔ ~ P ^ R

• ~ Q ↔ ~ P ^ R

Let S be a non empty subset of R. Consider the
following statement:
P: There is a rational number x∈S such that x > 0.
Which of the following statements is the negation of the statement P?

• There is a rational number x∈S such that x ≤ 0.

• There is no rational number x∈ S such that x≤0.

• Every rational number x∈S satisfies x ≤ 0.

• Every rational number x∈S satisfies x ≤ 0.

8.

The following statement
(p → q ) → [(~p → q) → q] is

• a fallacy

• a tautology

• equivalent to ~ p → q

• equivalent to ~ p → q

The remainder left out when 8²ⁿ –(62)²ⁿ⁺¹ is divided by 9 is 

• 0

• 2

• 7

• 7

Statement 1: ~ (p ↔ ~ q) is equivalent to p ↔ q
Statement 2 : ~ (p ↔ ~ q) is a tautology

• Statement–1 is true, Statement–2 is true, Statement–2 is a correct explanation for statement–1

• Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for statement–1.

• Statement–1 is true, statement–2 is false.

• Statement–1 is true, statement–2 is false.