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
pqp ∨ (q → p)11100100

Exactly one row of the truth table makes p ∨ (q → p) false. Click that row.

p ∨ (q → p)
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
2.Easy
ab¬a → ¬b11100100

Exactly one row of the truth table makes ¬a → ¬b false. Click that row.

¬a → ¬b
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
3.Medium
c ∧ ¬bc ↔ ¬b¬c ↔ ¬b¬c ∨ ¬b

Which formula is logically equivalent to ¬(c ↔ b)? Click it.

¬(c ↔ b)
  1. 1c ∧ ¬b
  2. 2c ↔ ¬b
  3. 3¬c ↔ ¬b
  4. 4¬c ∨ ¬b
4.Medium
¬¬q → ¬(r ∧ p)¬q → (r ∧ p)(r ∧ p) ∧ ¬¬q¬(r ∧ p) → ¬¬q

You want to prove (r ∧ p) → ¬q by contrapositive. Which implication do you prove instead? Click it.

(r ∧ p) → ¬q
  1. 1¬¬q → ¬(r ∧ p)
  2. 2¬q → (r ∧ p)
  3. 3(r ∧ p) ∧ ¬¬q
  4. 4¬(r ∧ p) → ¬¬q
5.Hard
∀x (P(x) ∧ ¬Q(x))∀x (P(x) → ¬Q(x))∃x (P(x) ∧ ¬Q(x))∀x (¬P(x) → ¬Q(x))

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

¬∃x (P(x) → Q(x))
  1. 1∀x (P(x) ∧ ¬Q(x))
  2. 2∀x (P(x) → ¬Q(x))
  3. 3∃x (P(x) ∧ ¬Q(x))
  4. 4∀x (¬P(x) → ¬Q(x))
6.Hard

How many of the 16 rows of the truth table make ((q ∧ s) ↔ (p ↔ r)) ↔ ¬((s ∨ p) ∨ ¬q) true? Call this number N.

((q ∧ s) ↔ (p ↔ r)) ↔ ¬((s ∨ p) ∨ ¬q)
N =

Answer key

  1. 1. p = 0, q = 1
  2. 2. a = 0, b = 1
  3. 3. c ↔ ¬b
  4. 4. ¬¬q → ¬(r ∧ p)
  5. 5. ∀x (P(x) ∧ ¬Q(x))
  6. 6. N = 8

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