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What a Truth Table Shows

A free Logic lesson from the “Truth Tables” unit, with a worked example and practice problems including step-by-step solutions.

List every truth assignment systematically. For two variables use TT, TF, FT, FF. Add an intermediate column for each subexpression, then compute the outer connective. Implication is false only at true antecedent and false consequent; a biconditional is true when the values agree.

What you'll learn

Why it matters: Truth tables are the bridge from language to Boolean thinking in computer science, circuits, spreadsheets, and careful casework.

Worked example

Problem. Complete the final column for ¬p ∨ q. Row order (p,q): TT, TF, FT, FF.

  1. At T, T, ¬p ∨ q is true.
  2. At T, F, ¬p ∨ q is false.
  3. At F, T, ¬p ∨ q is true.
  4. At F, F, ¬p ∨ q is true.

Answer: T, F, T, T

Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Complete the final column for ¬p ∨ q. Row order (p,q): TT, TF, FT, FF.

Choices: T, F, T, T · F, T, F, F · F, F, T, T · T, F, T, F

Show solution
  1. At T, T, ¬p ∨ q is true.
  2. At T, F, ¬p ∨ q is false.
  3. At F, T, ¬p ∨ q is true.
  4. At F, F, ¬p ∨ q is true.

Answer: T, F, T, T

2. Complete the final column for p ∧ ¬q. Row order (p,q): TT, TF, FT, FF.

Choices: T, F, T, T · T, T, F, F · F, T, F, T · F, T, F, F

Show solution
  1. At T, T, p ∧ ¬q is false.
  2. At T, F, p ∧ ¬q is true.
  3. At F, T, p ∧ ¬q is false.
  4. At F, F, p ∧ ¬q is false.

Answer: F, T, F, F

3. Complete the final column for p ↔ q. Row order (p,q): TT, TF, FT, FF.

Choices: F, F, F, T · T, F, F, F · T, F, F, T · F, T, T, F

Show solution
  1. At T, T, p ↔ q is true.
  2. At T, F, p ↔ q is false.
  3. At F, T, p ↔ q is false.
  4. At F, F, p ↔ q is true.

Answer: T, F, F, T

4. Complete the final column for ¬(p ∨ q). Row order (p,q): TT, TF, FT, FF.

Choices: F, F, F, F · F, F, F, T · T, T, T, F · T, F, F, T

Show solution
  1. At T, T, ¬(p ∨ q) is false.
  2. At T, F, ¬(p ∨ q) is false.
  3. At F, T, ¬(p ∨ q) is false.
  4. At F, F, ¬(p ∨ q) is true.

Answer: F, F, F, T

5. The final column in row order TT, TF, FT, FF is T, F, F, F. Which connective matches?

Choices: p ∧ q · p ∨ q · p → q · p ↔ q

Show solution
  1. Only the both-true row succeeds, which is conjunction.

Answer: p ∧ q

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