Identify the quantifier in the following statement and write the negation of the statement.
For every real number $x, x$ is less than $x+1.$

The simplest form of the following switching circuit is:

For each of the following compound statements,first identify the connecting words and then break it into component statements.
$x=2$ and $x=3$ are the roots of the equation $3x^{2} - x - 10 = 0$.

If $p: 25$ is an odd prime number.
$q: 14$ is a composite number and
$r: 64$ is a perfect square number.
Then which of the following statement patterns is true?

Consider the following two statements :
$P :$ If $7$ is an odd number,then $7$ is divisible by $2.$
$Q :$ If $7$ is a prime number,then $7$ is an odd number.
If $V_1$ is the truth value of the contrapositive of $P$ and $V_2$ is the truth value of the contrapositive of $Q,$ then the ordered pair $(V_1, V_2)$ equals

$(p \wedge r) \Leftrightarrow (p \wedge (\sim q))$ is equivalent to $(\sim p)$ when $r$ is.