Q.State which of the following are statements. For those that are, give the truth value.

Translate each symbol: $p$ = "He swims", $\sim q$ = "Water is not warm", and $\leftrightarrow$ = "if and only if". Combining gives "He swims if and only if water is not warm."

We are given:

• $p$ : He swims
• $q$ : Water is warm

Break down the symbolic statement $p \leftrightarrow \sim q$:

• $\sim q$ is the negation of $q$, i.e. "Water is not warm".
• The connective $\leftrightarrow$ is the biconditional, verbalised as "if and only if".

Putting the pieces together, $p \leftrightarrow \sim q$ reads:

$q$ = "Water is warm", $p$ = "He swims", and $\rightarrow$ = "if ... then ...". So $q \rightarrow p$ reads "If water is warm, then he swims."

We are given:

• $p$ : He swims
• $q$ : Water is warm

For the symbolic statement $q \rightarrow p$:

• The connective $\rightarrow$ is the conditional (implication), verbalised as "If ... then ...".
• The statement written before the arrow, $q$, is the hypothesis ("Water is warm"); the statement after the arrow, $p$, is the conclusion ("He swims").

Therefore the verbal statement is:

A statement must be a declarative sentence with a definite truth value. "Come here" is a command, so it is not a statement.

Test each option against the definition of a statement:

• "Smoking is injurious to health" — a declarative sentence (true); it is a statement.
• $2 + 2 = 4$ — a declarative sentence (true); it is a statement.
• "2 is the only even prime number." — a declarative sentence (true); it is a statement.
• "Come here" — an imperative sentence (a command). It is neither true nor false, so it has no truth value.

Only the last option lacks a truth value.

Swapping the two parts of the implication $q \rightarrow p$ gives its converse $p \rightarrow q$.

For a conditional statement of the form $\alpha \rightarrow \beta$ (read "if $\alpha$ then $\beta$"), the converse is obtained by interchanging the hypothesis and the conclusion, producing $\beta \rightarrow \alpha$.

Here the given statement is $q \rightarrow p$, so $\alpha = q$ and $\beta = p$. Interchanging them:

$q \rightarrow p \quad\Longrightarrow\quad p \rightarrow q.$