Truth Tables
Logical Deduction: Truth Tables
Truth Tables
Truth Tables
What you'll learn
- How truth tables list all combinations of truth values for propositions and compound statements.
- To evaluate AND (∧), OR (∨), NOT (¬), and IF-THEN (→) using standard rules.
- To build tables for two and three variables and identify tautologies and contradictions.
- To connect truth tables to validity of arguments in Class 11 logical deduction.
Key concepts
Level 1 — Foundations
Verbal: Each proposition is True (T) or False (F). A truth table shows the output of a logical connective for every input combination.
Basic connectives (A, B propositions):
| A | B | A∧B | A∨B | ¬A | A→B |
|---|---|---|---|---|---|
| T | T | T | T | F | T |
| T | F | F | T | F | F |
| F | T | F | T | T | T |
| F | F | F | F | T | T |
Key: A→B is F only when A is T and B is F (true premise, false conclusion). Otherwise T.
Three variables: 2³ = 8 rows. Systematic order: TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
Tautology: Column always T. Contradiction: Column always F.
Level 2 — Exam depth
Biconditional A↔B: T when A and B match; build as (A→B)∧(B→A).
De Morgan in tables: ¬(A∧B) same column as ¬A∨¬B — verify on 4 rows for proof.
Short-cut for validity: Argument with premises P1, P2 and conclusion C is valid if there is no row where all premises are T and conclusion F.
Real statements: "If it rains, the match is cancelled" — R→M. Test row R=T, M=F → implication F → counterexample to universal claim.
Exam tip: For 2 variables, 4 rows fit on one line of rough work; circle rows violating a rule.
Worked example
Build truth table for (P∨Q)∧¬P
P | Q | P∨Q | ¬P | (P∨Q)∧¬P
T | T | T | F | F
T | F | T | F | F
F | T | T | T | T
F | F | F | T | F
Only row F,T yields T → equivalent to **¬P∧Q** (check mentally).
Test validity: premises P→Q, P; conclusion Q
Combined: look for row with P→Q=T, P=T, Q=F.
Only F,F gives P→Q=T with P=F — no row has P=T and Q=F with P→Q=T.
**Valid argument (Modus Ponens).**
Common mistakes
| Mistake | Why it happens | Fix |
|---|---|---|
| A→B false when B is false | Confused with B→A | Only F when A=T and B=F |
| Missing rows in 3-variable table | Counted wrong | Exactly 2ⁿ rows for n variables |
| OR vs XOR | Exclusive or assumed | ∨ is inclusive unless stated 'either but not both' |
| Calling satisfiable formula a tautology | One T row enough | Tautology = all rows T |
Quick check
- How many rows for 4 propositions?
- Show one row where A→B and B→A differ.
- Is (P∨¬P) a tautology? One-line justification.
- Stretch: Write argument form for "denying the antecedent" and show invalid row.
Revision tip: Revisit adjacent topics in Logical Deduction before mixed practice on Truth Tables.
Open the Practice tab for graded questions on Truth Tables.
Exam strategy
For validity questions, build the table with columns for each premise and the conclusion — scan for the forbidden row (premises T, conclusion F). Memorise the conditional's single false row to avoid rebuilding from scratch. When pressed for time on 3-variable tables, test only rows where the conclusion is F — halve the work. Label columns P, Q, R clearly; misaligned columns cause catastrophic errors.
Key Takeaways (TL;DR)
- What you'll learn
- Key concepts
- Worked example
- Common mistakes
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