Q.Write the negations of the following statements:

The negation of a statement $p$ is "not $p$" — the direct logical opposite of what $p$ claims.

(i) Original statement: "Both the diagonals of a rectangle have the same length."

Negation: "Both the diagonals of a rectangle do not have the same length."

(ii) Original statement: "$\sqrt{7}$ is a rational number."

Negation: "$\sqrt{7}$ is not a rational number."

…

Negate a universal ("all") statement with "some ... not"; negate a disjunction $p \vee q$ with $\sim p \wedge \sim q$ (De Morgan's law).

(a) The statement "All students of this college live in the hostel" is a universal statement of the form "For all $x$, $p(x)$". Its negation is "There exists a student who does not live in the hostel", i.e.

$\text{Some students of this college do not live in the hostel.}$

(b) The statement "6 is an even number or 36 is a perfect square" has the form $p \vee q$, where

$p$: 6 is an even number, $q$: 36 is a perfect square.

…

The negation of a statement $p$, written $\sim p$, is true exactly when $p$ is false; here it is formed by inserting "not" into each statement.

(i) Statement: "Shillong is the capital of Kerela."

Negation: "Shillong is not the capital of Kerela." (Equivalently: "It is false that Shillong is the capital of Kerela.")

(ii) Statement: "3 is greater than 5."

Negation: "3 is not greater than 5." (Equivalently: "It is false that 3 is greater than 5.")

…