Propositional logic & proofs worksheet with answers

Free printable Propositional logic & proofs worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy

How many rows N does the complete truth table of (a ↔ ¬c) ↔ (a ↔ (¬c → b)) have?

(a ↔ ¬c) ↔ (a ↔ (¬c → b))
N =
2.Easy
pq¬q ∧ (p ↔ q)11100100

Exactly one row of the truth table makes ¬q ∧ (p ↔ q) true. Click that row.

¬q ∧ (p ↔ q)
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
3.Medium
(b ∨ c) ∧ a(b ∧ c) ∨ (b ∧ a)b ∧ (c ∨ a)(b ∨ c) ∧ (b ∨ a)

Which formula is logically equivalent to b ∨ (c ∧ a)? Click it.

b ∨ (c ∧ a)
  1. 1(b ∨ c) ∧ a
  2. 2(b ∧ c) ∨ (b ∧ a)
  3. 3b ∧ (c ∨ a)
  4. 4(b ∨ c) ∧ (b ∨ a)
4.Medium
tautologycontradictioncontingency

Is ¬(q ∨ ¬q) a tautology, a contradiction or a contingency? Click your answer.

¬(q ∨ ¬q)
  1. 1tautology
  2. 2contradiction
  3. 3contingency
5.Hard
pqrr → (p ∨ q)rp111110101100011010001000

The argument with premises r → (p ∨ q); r and conclusion p is invalid. Click the row of the truth table that is a counterexample.

r → (p ∨ q)r∴ p
  1. 1p = 1, q = 1, r = 1
  2. 2p = 1, q = 1, r = 0
  3. 3p = 1, q = 0, r = 1
  4. 4p = 1, q = 0, r = 0
  5. 5p = 0, q = 1, r = 1
  6. 6p = 0, q = 1, r = 0
  7. 7p = 0, q = 0, r = 1
  8. 8p = 0, q = 0, r = 0
6.Hard
∃x ¬P(x)∃x P(x)∀x P(x)∀x ¬P(x)

Which statement is equivalent to ¬∀x P(x), with the negation moved all the way inside? Click it.

¬∀x P(x)
  1. 1∃x ¬P(x)
  2. 2∃x P(x)
  3. 3∀x P(x)
  4. 4∀x ¬P(x)

Answer key

  1. 1. N = 8
  2. 2. p = 0, q = 0
  3. 3. (b ∨ c) ∧ (b ∨ a)
  4. 4. contradiction
  5. 5. p = 0, q = 1, r = 1
  6. 6. ∃x ¬P(x)

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