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
ab¬((a ∨ b) ∧ (b ∧ a))11?10?01?00?

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

¬((a ∨ b) ∧ (b ∧ a))
N =
2.Easy
pqq → ¬p11100100

Exactly one row of the truth table makes q → ¬p false. Click that row.

q → ¬p
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
3.Medium
¬c ∧ a¬c ∧ ¬a¬c ∨ ¬a¬c ∨ a

Which formula is logically equivalent to ¬(c ∧ a)? Click it.

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

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

r → (p ∨ q)
  1. 1(p ∨ q) → r
  2. 2r ∧ ¬(p ∨ q)
  3. 3¬(p ∨ q) → ¬r
  4. 4¬r → ¬(p ∨ q)
5.Hard
prr ∨ pr¬p11100100

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

r ∨ pr∴ ¬p
  1. 1p = 1, r = 1
  2. 2p = 1, r = 0
  3. 3p = 0, r = 1
  4. 4p = 0, r = 0
6.Hard
∃x ∃y (R(x, y) ∧ ¬R(y, x))∃x ∃y (¬R(x, y) ∧ ¬R(y, x))∀x ∀y (R(x, y) ∧ ¬R(y, x))∃x ∃y (R(x, y) → ¬R(y, x))

Which statement is equivalent to ¬∀x ∀y (R(x, y) → R(y, x)), with the negation moved all the way inside? Click it.

¬∀x ∀y (R(x, y) → R(y, x))
  1. 1∃x ∃y (R(x, y) ∧ ¬R(y, x))
  2. 2∃x ∃y (¬R(x, y) ∧ ¬R(y, x))
  3. 3∀x ∀y (R(x, y) ∧ ¬R(y, x))
  4. 4∃x ∃y (R(x, y) → ¬R(y, x))

Answer key

  1. 1. N = 3
  2. 2. p = 1, q = 1
  3. 3. ¬c ∨ ¬a
  4. 4. ¬(p ∨ q) → ¬r
  5. 5. p = 1, r = 1
  6. 6. ∃x ∃y (R(x, y) ∧ ¬R(y, x))

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