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 ↔ q11?10?01?00?

Complete the truth table (1 = true, 0 = false). In how many rows N is p ↔ q true?

p ↔ q
N =
2.Easy
ab(¬b ↔ ¬a) ∨ (a ∧ ¬b)11?10?01?00?

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

(¬b ↔ ¬a) ∨ (a ∧ ¬b)
N =
3.Medium
b → (c ∨ a)¬b → ¬(c ∨ a)¬(c ∨ a) → ¬b(c ∨ a) ∧ ¬b

You want to prove (c ∨ a) → b by contrapositive. Which implication do you prove instead? Click it.

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

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

(p ∨ r) → q
  1. 1p ∨ (r → q)
  2. 2(p → q) ∨ (r → q)
  3. 3(p → q) ∧ (r → q)
  4. 4(p ∧ r) → q
5.Hard
prp → rrp11100100

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

p → rr∴ p
  1. 1p = 1, r = 1
  2. 2p = 1, r = 0
  3. 3p = 0, r = 1
  4. 4p = 0, r = 0
6.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))

Answer key

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

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