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

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

¬((r ∧ (q → p)) ∨ ((q ∧ ¬p) → ¬r))
N =
3.Medium
¬p → ¬(q ∧ r)p → (q ∧ r)(q ∧ r) ∧ ¬p¬(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¬p → ¬(q ∧ r)
  2. 2p → (q ∧ r)
  3. 3(q ∧ r) ∧ ¬p
  4. 4¬(q ∧ r) → ¬p
4.Medium
a → c¬c ∨ ac ∨ ¬a¬c → ¬a

Which formula is logically equivalent to c → a? Click it.

c → a
  1. 1a → c
  2. 2¬c ∨ a
  3. 3c ∨ ¬a
  4. 4¬c → ¬a
5.Hard
pqrr → (p ∨ q)rp111110101100011010001000

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

r → (p ∨ q)r∴ p
  1. 1p = 1, q = 1, r = 1
  2. 2p = 1, q = 1, r = 0
  3. 3p = 1, q = 0, r = 1
  4. 4p = 1, q = 0, r = 0
  5. 5p = 0, q = 1, r = 1
  6. 6p = 0, q = 1, r = 0
  7. 7p = 0, q = 0, r = 1
  8. 8p = 0, q = 0, r = 0
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 = 1, q = 1
  2. 2. N = 8
  3. 3. ¬p → ¬(q ∧ r)
  4. 4. ¬c ∨ a
  5. 5. p = 0, q = 1, r = 1
  6. 6. ∀x (F(x) ∧ ¬G(x))

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