Truth Tables with Three Variables
A free Logic lesson from the “Truth Tables” unit, with a worked example and practice problems including step-by-step solutions.
Three independent Boolean variables give eight rows. In TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF order, evaluate parenthesized parts first. To disprove equivalence it is enough to find one row on which the final columns differ; to establish equivalence every row must agree.
What you'll learn
- Build the eight-row table for three variables
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Complete the final column for (p ∧ q) ∨ r. Row order (p,q,r): TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
- At T, T, T, (p ∧ q) ∨ r is true.
- At T, T, F, (p ∧ q) ∨ r is true.
- At T, F, T, (p ∧ q) ∨ r is true.
- At T, F, F, (p ∧ q) ∨ r is false.
- At F, T, T, (p ∧ q) ∨ r is true.
- At F, T, F, (p ∧ q) ∨ r is false.
- At F, F, T, (p ∧ q) ∨ r is true.
- At F, F, F, (p ∧ q) ∨ r is false.
Answer: T, T, T, F, T, F, T, F
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) ∨ r. Row order (p,q,r): TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
Choices: F, T, T, F, T, F, T, F · T, T, T, F, T, F, T, T · T, T, T, F, T, F, T, F · F, F, F, T, F, T, F, T
Show solution
- At T, T, T, (p ∧ q) ∨ r is true.
- At T, T, F, (p ∧ q) ∨ r is true.
- At T, F, T, (p ∧ q) ∨ r is true.
- At T, F, F, (p ∧ q) ∨ r is false.
- At F, T, T, (p ∧ q) ∨ r is true.
- At F, T, F, (p ∧ q) ∨ r is false.
- At F, F, T, (p ∧ q) ∨ r is true.
- At F, F, F, (p ∧ q) ∨ r is false.
Answer: T, T, T, F, T, F, T, F
2. Complete the final column for p ∧ (q ∨ r). Row order (p,q,r): TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
Choices: T, T, T, F, F, F, F, T · T, T, T, F, F, F, F, F · F, F, F, T, T, T, T, T · F, T, T, F, F, F, F, F
Show solution
- At T, T, T, p ∧ (q ∨ r) is true.
- At T, T, F, p ∧ (q ∨ r) is true.
- At T, F, T, p ∧ (q ∨ r) is true.
- At T, F, F, p ∧ (q ∨ r) is false.
- At F, T, T, p ∧ (q ∨ r) is false.
- At F, T, F, p ∧ (q ∨ r) is false.
- At F, F, T, p ∧ (q ∨ r) is false.
- At F, F, F, p ∧ (q ∨ r) is false.
Answer: T, T, T, F, F, F, F, F
3. Complete the final column for (p → q) ∧ r. Row order (p,q,r): TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
Choices: T, F, F, F, T, F, T, F · F, T, T, T, F, T, F, T · F, F, F, F, T, F, T, F · T, F, F, F, T, F, T, T
Show solution
- At T, T, T, (p → q) ∧ r is true.
- At T, T, F, (p → q) ∧ r is false.
- At T, F, T, (p → q) ∧ r is false.
- At T, F, F, (p → q) ∧ r is false.
- At F, T, T, (p → q) ∧ r is true.
- At F, T, F, (p → q) ∧ r is false.
- At F, F, T, (p → q) ∧ r is true.
- At F, F, F, (p → q) ∧ r is false.
Answer: T, F, F, F, T, F, T, F
4. Complete the final column for ¬p ∨ (q ∧ r). Row order (p,q,r): TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF.
Choices: F, T, T, T, F, F, F, F · F, F, F, F, T, T, T, T · T, F, F, F, T, T, T, F · T, F, F, F, T, T, T, T
Show solution
- At T, T, T, ¬p ∨ (q ∧ r) is true.
- At T, T, F, ¬p ∨ (q ∧ r) is false.
- At T, F, T, ¬p ∨ (q ∧ r) is false.
- At T, F, F, ¬p ∨ (q ∧ r) is false.
- At F, T, T, ¬p ∨ (q ∧ r) is true.
- At F, T, F, ¬p ∨ (q ∧ r) is true.
- At F, F, T, ¬p ∨ (q ∧ r) is true.
- At F, F, F, ¬p ∨ (q ∧ r) is true.
Answer: T, F, F, F, T, T, T, T
5. At p=false, q=true, r=true, compare (p ∧ q) ∨ r with p ∧ (q ∨ r).
Choices: Both false · The first is false and the second true · The first is true and the second false · Both true
Show solution
- The first becomes (false AND true) OR true = true.
- The second begins with a false conjunction factor, so is false.
Answer: The first is true and the second false
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Practice Truth Tables with Three Variables Take the free placement check