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
pqr¬(¬r ∨ ¬q) ∨ (r → p)111?110?101?100?011?010?001?000?

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

¬(¬r ∨ ¬q) ∨ (r → p)
N =
2.Easy
pqp ∨ (q → p)11?10?01?00?

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

p ∨ (q → p)
N =
3.Medium
abc¬a ∧ c(a → b) → (¬a ∧ c)¬(c ↔ b) → ((a → b) → (¬a ∧ c))111?110?101?100?011?010?001?000?

In how many of the 8 rows of its truth table is ¬(c ↔ b) → ((a → b) → (¬a ∧ c)) true? Call this number N.

¬(c ↔ b) → ((a → b) → (¬a ∧ c))
N =
4.Medium
(p ∧ r) → ¬q¬(p ∧ r) → ¬¬q¬q ∧ ¬(p ∧ r)¬¬q → ¬(p ∧ r)

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

¬q → (p ∧ r)
  1. 1(p ∧ r) → ¬q
  2. 2¬(p ∧ r) → ¬¬q
  3. 3¬q ∧ ¬(p ∧ r)
  4. 4¬¬q → ¬(p ∧ r)
5.Hard
∃y ∀x ¬R(x, y)∀x ∃y ¬R(x, y)∀x ∀y ¬R(x, y)∃x ∀y ¬R(x, y)

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

¬∀x ∃y R(x, y)
  1. 1∃y ∀x ¬R(x, y)
  2. 2∀x ∃y ¬R(x, y)
  3. 3∀x ∀y ¬R(x, y)
  4. 4∃x ∀y ¬R(x, y)
6.Hard
pqrr → qp → qr → p111110101100011010001000

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

r → qp → 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

Answer key

  1. 1. N = 7
  2. 2. N = 3
  3. 3. N = 6
  4. 4. ¬(p ∧ r) → ¬¬q
  5. 5. ∃x ∀y ¬R(x, y)
  6. 6. p = 0, q = 1, r = 1

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