For two different persons $x$ and $y$, the predicate $M(x, y)$ denotes that $x$ knows $y$. Consider the following statement.

There is a person who does not know anyone else, but that person is known by everyone else.

Which one of the following expressions represents the above statement?

Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

Let p and q be the following propositions:

p: Fail grade can be given.

q: Student scores more than 50% marks.

Consider the statement: “Fail grade cannot be given when student scores more than 50% marks.”

Which one of the following is the CORRECT representation of the above statement in propositional logic?

Geetha has a conjecture about integers, which is of the form

$$\forall x\left( {P(x) \Rightarrow \exists yQ(x,y)} \right)$$,

where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha's conjecture?