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
pqq → ¬p11?10?01?00?

Complete the truth table (1 = true, 0 = false). In how many rows N is q → ¬p true?

q → ¬p
N =
2.Easy
ab¬(a ∧ ¬b) → a11?10?01?00?

Complete the truth table (1 = true, 0 = false). In how many rows N is ¬(a ∧ ¬b) → a true?

¬(a ∧ ¬b) → a
N =
3.Medium
tautologycontradictioncontingency

Is b ↔ ¬b a tautology, a contradiction or a contingency? Click your answer.

b ↔ ¬b
  1. 1tautology
  2. 2contradiction
  3. 3contingency
4.Medium
p ↔ ¬q¬p ∨ ¬qp ∧ ¬q¬p ↔ ¬q

Which formula is logically equivalent to ¬(p ↔ q)? Click it.

¬(p ↔ q)
  1. 1p ↔ ¬q
  2. 2¬p ∨ ¬q
  3. 3p ∧ ¬q
  4. 4¬p ↔ ¬q
5.Hard
∀x ∀y ¬R(x, y)∃x ∃y ¬R(x, y)∃x ∀y ¬R(x, y)∀x ∃y ¬R(x, y)

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

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

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

¬∀x (F(x) → G(x))
  1. 1∃x (F(x) → ¬G(x))
  2. 2∃x (F(x) ∧ ¬G(x))
  3. 3∃x (¬F(x) ∧ G(x))
  4. 4∀x (F(x) ∧ ¬G(x))

Answer key

  1. 1. N = 3
  2. 2. N = 2
  3. 3. contradiction
  4. 4. p ↔ ¬q
  5. 5. ∀x ∃y ¬R(x, y)
  6. 6. ∃x (F(x) ∧ ¬G(x))

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