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 → (b ∧ ¬a)11100100

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

b → (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¬((q → p) ∨ p)11100100

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

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

Which formula is logically equivalent to q ↔ r? Click it.

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

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

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

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