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
ab¬((a ∨ ¬b) → ¬b)11?10?01?00?

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

¬((a ∨ ¬b) → ¬b)
N =
2.Easy
pq(q ↔ p) ∧ p11100100

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
p ∨ (r ∧ q)(p ∧ r) ∨ (p ∧ q)(p ∧ r) ∨ q(p ∨ r) ∧ (p ∨ q)

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

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

You want to prove (a ∨ b) → c by contradiction. Which statement do you assume at the start? Click it.

(a ∨ b) → c
  1. 1¬(a ∨ b) → ¬c
  2. 2¬(a ∨ b) ∧ ¬c
  3. 3(a ∨ b) ∧ ¬c
  4. 4¬(a ∨ b) ∧ c
5.Hard
∃x (¬A(x) ∨ ¬B(x))∃x (¬A(x) ∧ ¬B(x))∀x (¬A(x) ∨ ¬B(x))∀x (¬A(x) ∧ ¬B(x))

Which statement is equivalent to ¬∀x (A(x) ∨ B(x)), with the negation moved all the way inside? Click it.

¬∀x (A(x) ∨ B(x))
  1. 1∃x (¬A(x) ∨ ¬B(x))
  2. 2∃x (¬A(x) ∧ ¬B(x))
  3. 3∀x (¬A(x) ∨ ¬B(x))
  4. 4∀x (¬A(x) ∧ ¬B(x))
6.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))

Answer key

  1. 1. N = 1
  2. 2. p = 1, q = 1
  3. 3. (p ∧ r) ∨ (p ∧ q)
  4. 4. (a ∨ b) ∧ ¬c
  5. 5. ∃x (¬A(x) ∧ ¬B(x))
  6. 6. ∃x (A(x) ∧ ∀y ¬R(x, y))

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