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 → ¬q) ↔ p)11100100

Exactly one row of the truth table makes ¬((p → ¬q) ↔ p) false. 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
pqq → (p ↔ q)11100100

Exactly one row of the truth table makes q → (p ↔ q) false. Click that row.

q → (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) → ¬q¬¬q → ¬(r ∨ p)

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

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

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

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

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

((a → d) ∨ d) → (c ∨ (b ↔ c))
N =

Answer key

  1. 1. p = 1, q = 0
  2. 2. p = 0, q = 1
  3. 3. ¬(r ∨ p) → ¬¬q
  4. 4. ¬(q ∨ p) → ¬r
  5. 5. ∃x (¬A(x) ∧ ¬B(x))
  6. 6. N = 13

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