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

How many rows N does the complete truth table of ¬((r ↔ ¬q) ∨ ¬r) ∧ (¬q ↔ ¬p) have?

¬((r ↔ ¬q) ∨ ¬r) ∧ (¬q ↔ ¬p)
N =
2.Easy
abb ∧ ¬a11100100

Exactly one row of the truth table makes b ∧ ¬a true. 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
3.Medium
¬q → ¬r¬r → ¬q¬q ∧ ¬rq → r

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

r → q
  1. 1¬q → ¬r
  2. 2¬r → ¬q
  3. 3¬q ∧ ¬r
  4. 4q → r
4.Medium
tautologycontradictioncontingency

Is r ∧ ¬r a tautology, a contradiction or a contingency? Click your answer.

r ∧ ¬r
  1. 1tautology
  2. 2contradiction
  3. 3contingency
5.Hard
pqrr → qp → qr → p111110101100011010001000

The argument with premises r → q; p → q and conclusion r → p is invalid. Click the row of the truth table that is a counterexample.

r → qp → q∴ r → p
  1. 1p = 1, q = 1, r = 1
  2. 2p = 1, q = 1, r = 0
  3. 3p = 1, q = 0, r = 1
  4. 4p = 1, q = 0, r = 0
  5. 5p = 0, q = 1, r = 1
  6. 6p = 0, q = 1, r = 0
  7. 7p = 0, q = 0, r = 1
  8. 8p = 0, q = 0, r = 0
6.Hard
∃x (F(x) → ∀y ¬R(x, y))∃x (F(x) ∧ ∀y ¬R(x, y))∀x (F(x) ∧ ∀y ¬R(x, y))∃x (¬F(x) ∧ ∃y R(x, y))

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

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

Answer key

  1. 1. N = 8
  2. 2. a = 0, b = 1
  3. 3. ¬q → ¬r
  4. 4. contradiction
  5. 5. p = 0, q = 1, r = 1
  6. 6. ∃x (F(x) ∧ ∀y ¬R(x, y))

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