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
pqp ∧ (q ↔ ¬p)11100100

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

p ∧ (q ↔ ¬p)
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
2.Easy
pqp ∨ q11100100

Exactly one row of the truth table makes p ∨ q false. 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
3.Medium
¬q → ¬(r ∨ p)¬(r ∨ p) → ¬q(r ∨ p) ∧ ¬qq → (r ∨ p)

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

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

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

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

How many of the 16 rows of the truth table make ((¬d ∨ b) ∧ (c → b)) ↔ ((d ∨ c) ∨ ¬a) true? Call this number N.

((¬d ∨ b) ∧ (c → b)) ↔ ((d ∨ c) ∨ ¬a)
N =

Answer key

  1. 1. p = 1, q = 0
  2. 2. p = 0, q = 0
  3. 3. ¬q → ¬(r ∨ p)
  4. 4. (r ∨ p) ∧ (r ∨ q)
  5. 5. ∀x (P(x) ∧ ¬Q(x))
  6. 6. N = 8

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