Mathematical Reasoning • Topic 2 of 3

Truth Tables

A truth table lists every combination of truth values for the simple statements and records the result for the compound one. With two variables there are four rows (TT, TF, FT, FF); with three there are eight (2ⁿ rows for n variables). The key behaviours to memorise: ¬p just flips p; p ∧ q is true only in the all-true row; p ∨ q is false only in the all-false row; p → q is false ONLY when p is true and q is false (a true premise leading to a false conclusion); p ↔ q is true exactly when p and q match. A compound that comes out true in every row is a tautology; one false in every row is a contradiction. In CAT and XAT you rarely draw the full table — you scan for the single damning row. To prove an implication false you only need one case where the premise holds but the conclusion fails; to prove a tautology you confirm no such failing row can exist.

✅ Solved examples

1. Build the truth table for p → q and state when it is false.
Rows: TT→T, TF→F, FT→T, FF→T. So p → q is false ONLY when p is true and q is false. A false premise (last two rows) makes the implication vacuously true.
2. Is (p ∧ ¬p) a tautology, a contradiction, or neither?
p and ¬p can never both be true, so p ∧ ¬p is false in every row — it is a contradiction.
3. Is (p ∨ ¬p) a tautology?
In every row exactly one of p, ¬p is true, so the OR is true in all rows — yes, a tautology (the law of excluded middle).
4. Verify p → q ≡ ¬p ∨ q using a truth table.
p → q gives T,F,T,T. ¬p ∨ q gives: ¬p = F,F,T,T so ¬p ∨ q = T,F,T,T. The columns match in all four rows, so they are logically equivalent.

✏️ Practice — try these, take hints as needed

1. How many rows does a truth table for 4 distinct simple statements have?
Each variable is T or F.
Combinations multiply.
2 to the power of n.
16 rows.
2. In which single row is p ∨ q false?
OR needs at least one true.
Only fails when none is true.
Both p and q false.
When both p and q are false (FF).
3. Is (p → q) ∧ (q → p) equivalent to p ↔ q?
Biconditional means both directions.
p → q and q → p together.
Check each row matches p ↔ q.
Yes — they are logically equivalent.
4. Classify ¬(p → q) ∨ (p → q).
It is A ∨ ¬A in disguise.
A statement OR its negation.
Always true.
Tautology.
5. For p = T, q = F, r = T, evaluate (p ∧ q) → r.
Evaluate the premise first.
p ∧ q = T ∧ F = F.
A false premise makes → true.
True.

📝 Topic test — 8 questions

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