Q.Verify whether the following compound propositions are tautologies or contradictions or contingency.

A truth table for $n$ independent propositional variables always has $2^n$ rows; here there are 3 variables.

1. The statement $(p\vee q)\wedge(p\vee r)$ involves $3$ distinct propositional variables: $p,q,r$. …

$p\vee(\sim p)$ is a tautology (always true), unlike the other three statement forms.

1. A tautology is a compound statement that is true for every truth value assignment of its component statements.
2. Check $p\vee(\sim p)$: if $p$ is True, $p\vee(\sim p)=T\vee F=T$; if $p$ is False, $p\vee(\sim p)=F\vee T=T$. It is always true — a tautology.
3. Check $p\wedge(\sim p)$: if $p$ is True, this is $T\wedge F=F$; it is never true (a contradiction, not a tautology). …

Evaluating each compound statement with $p=T,q=F$ shows (i), (iii) and (iv) are true while (ii) is false.

1. Given $p=T$ and $q=F$, so $\sim p = F$ and $\sim q = T$.
2. (i) $p\vee q = T\vee F = T$ (disjunction is true if at least one disjunct is true).
3. (ii) $\sim p\vee q = F\vee F = F$.
4. (iii) $p\vee\sim q = T\vee T = T$. …

Statement (c) is the false one: a biconditional $p\leftrightarrow q$ is a tautology only for specially related (logically equivalent) statements, not for arbitrary $p,q$.

1. (a) A contradiction is, by definition, a statement whose truth table's final column is all F — this is a correct/true statement.
2. (b) The double-negation law $\sim(\sim p)\equiv p$ is a standard tautological equivalence — this is a correct/true statement.
3. (d) A tautology is, by definition, a statement whose truth table's final column is all T — this is a correct/true statement. …

With $p=T,\ q=F$: statements (i), (iii), (iv) are true; (ii) is false.

1. (i) $p\vee q = T\vee F=T$ — true.
2. (ii) $\sim p\vee q = F\vee F=F$ — false.
3. (iii) $p\vee(\sim q)=T\vee T=T$ — true.
4. (iv) $p\wedge(\sim q)=T\wedge T=T$ — true. …

Only $p\wedge(\sim p)$ is false under every possible truth assignment, making it the contradiction.

1. A contradiction (or fallacy) is a compound statement whose truth table has F in every row.
2. Check (a) $p\vee q$: truth table has T for most rows (e.g. $p=T$), so it is neither a tautology nor a contradiction — a contingency.
3. Check (b) $p\wedge q$: false when either is false, true when both are true — again a contingency, not always false.
4. Check (c) $p\vee(\sim p)$: for any truth value of $p$, one of $p,\sim p$ is true, so this is always true — a tautology (the law of excluded middle), the opposite of a contradiction. …