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
abc¬((a ∨ b) → (¬a ↔ c))111?110?101?100?011?010?001?000?

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

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

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

p ∧ (p → q)
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
3.Medium
¬r ∨ ¬qr ∧ ¬qr → ¬q¬r → ¬q

Which formula is logically equivalent to ¬(r → q)? Click it.

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

Which formula is logically equivalent to ¬(r ∨ p)? Click it.

¬(r ∨ p)
  1. 1r ∨ ¬p
  2. 2¬r ∨ ¬p
  3. 3¬r ∧ p
  4. 4¬r ∧ ¬p
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
∀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)

Answer key

  1. 1. N = 3
  2. 2. p = 1, q = 1
  3. 3. r ∧ ¬q
  4. 4. ¬r ∧ ¬p
  5. 5. ∃x (¬P(x) ∧ ¬Q(x))
  6. 6. ∀x ∃y ¬R(x, y)

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