Let $\Delta, \nabla \in \{\wedge, \vee\}$ be such that $(p \nabla q) \Rightarrow ((p \nabla q) \nabla r)$ is a tautology. Then $(p \nabla q) \Delta r$ is logically equivalent to

Are the following pairs of statements negations of each other:
$1$. The number $x$ is not a rational number.
$2$. The number $x$ is not an irrational number.

Which of the following statement$(s)$ is/are not true?
$I$) If $1$ is not a prime number, then $2$ is not a prime number.
$II$) $e$ is a vowel and $12 \times 3 = 36$.
$III$) It is not true that $14$ is a composite number and $3$ is an even number.
$IV$) $\sqrt{5}$ is an irrational number, but $3 + \sqrt{5}$ is a complex number.

The expression $\sim ( \sim p \to q)$ is logically equivalent to

Give three examples of sentences which are not statements. Give reasons for the answers.

Write the following statement in the form of "if-then":
The banana trees will bloom if it stays warm for a month.