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
pq¬p ∧ ¬q11100100

Exactly one row of the truth table makes ¬p ∧ ¬q true. Click that row.

¬p ∧ ¬q
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
2.Easy

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

¬(((r → q) ∧ p) ∨ (r ↔ (p ∧ ¬q)))
N =
3.Medium
(p → q) ∧ (q → p)p ∧ q(p → q) ∨ (q → p)(p → q) ∧ (¬q → ¬p)

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

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

You want to prove (p ∧ q) → ¬r by contrapositive. Which implication do you prove instead? Click it.

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

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