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.

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

• a fallacy

• a tautology

• equivalent to ~ p → q

• equivalent to ~ p → q

103.

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.

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”, and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p
Statement –2: r is equivalent to ∼ (p ↔ ∼ q).

• Statement −1 is false, Statement −2 is true

• 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 true; Statement −2 is not a correct explanation for Statement −1.

The statement p → (q → p) is equivalent to

• p → (p → q) 

• p → (p ∨ q)

• p → (p ∧ q)

• p → (p ∧ q)

Consider the following statements:
(a) Mode can be computed from histogram
(b) Median is not independent of change of scale
(c) Variance is independent of the change of origin and scale. Which of these is/are correct?

• only (a)

• only (b)

• only (a) and (b)

• only (a) and (b)

The Boolean expression ~ (p v q )v (-p ∧q ) is equivalent to

• ~q

• ~ p

• p

• q

If p, q, r are simple propositions with truth values T, F, T, then the truth value of $\left(~ \mathrm{p} \vee \mathrm{q}\right) \wedge ~ \mathrm{r} \Rightarrow \mathrm{p}$ is

• true

• false
  
  

• true, if r is false

• true, if q is true

If p : It rains today, q : I go to school, r : I shall meet any friends ands : I shall go for a movie, then which of the following is the proportion? If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie.

• $\left(~ \mathrm{p} \wedge ~ \mathrm{q}\right) \Rightarrow \left(\mathrm{r} \wedge \mathrm{s}\right)$

• $~ \left(\mathrm{p} \wedge \mathrm{q}\right) \Rightarrow \left(\mathrm{r} \wedge \mathrm{s}\right)$

• $~ \left(\mathrm{p} \vee \mathrm{q}\right) \Rightarrow \left(\mathrm{r} \wedge \mathrm{s}\right)$

• None of these