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
abb → a11100100

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

b → a
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
2.Easy
pq(p ↔ q) ∧ q11100100

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

(p ↔ q) ∧ q
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
3.Medium
tautologycontradictioncontingency

Is (a ∧ (a → b)) → b a tautology, a contradiction or a contingency? Click your answer.

(a ∧ (a → b)) → b
  1. 1tautology
  2. 2contradiction
  3. 3contingency
4.Medium
abc¬(¬b ∧ c)a → ¬(¬b ∧ c)(¬b ∧ a) → (a → ¬(¬b ∧ c))111?110?101?100?011?010?001?000?

In how many of the 8 rows of its truth table is (¬b ∧ a) → (a → ¬(¬b ∧ c)) true? Call this number N.

(¬b ∧ a) → (a → ¬(¬b ∧ c))
N =
5.Hard
∃x (¬A(x) ∧ ∃y R(x, y))∃x (A(x) → ∀y ¬R(x, y))∀x (A(x) ∧ ∀y ¬R(x, y))∃x (A(x) ∧ ∀y ¬R(x, y))

Which statement is equivalent to ¬∀x (A(x) → ∃y R(x, y)), with the negation moved all the way inside? Click it.

¬∀x (A(x) → ∃y R(x, y))
  1. 1∃x (¬A(x) ∧ ∃y R(x, y))
  2. 2∃x (A(x) → ∀y ¬R(x, y))
  3. 3∀x (A(x) ∧ ∀y ¬R(x, y))
  4. 4∃x (A(x) ∧ ∀y ¬R(x, y))
6.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))

Answer key

  1. 1. a = 0, b = 1
  2. 2. p = 1, q = 1
  3. 3. tautology
  4. 4. N = 7
  5. 5. ∃x (A(x) ∧ ∀y ¬R(x, y))
  6. 6. ∃x (¬P(x) ∧ ¬Q(x))

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