Q.If $p$ : He swims
$q$ : Water is warm
Give the verbal statement for the following symbolic statement:
$p \leftrightarrow \sim q$

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". …

$q$ = "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 ...". …

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. …

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 $\beta \rightarrow \alpha$.

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