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
ab(a ↔ b) ∨ (a ∧ ¬b)11?10?01?00?

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

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

In how many of the 8 rows of its truth table is (r → (p ∧ r)) ∧ (q ∧ (p ↔ q)) true? Call this number N.

(r → (p ∧ r)) ∧ (q ∧ (p ↔ q))
N =
4.Medium
pqr¬(r ∨ q)¬p ∧ ¬(r ∨ q)(q ∧ (¬r ∨ ¬p)) → (¬p ∧ ¬(r ∨ q))111?110?101?100?011?010?001?000?

In how many of the 8 rows of its truth table is (q ∧ (¬r ∨ ¬p)) → (¬p ∧ ¬(r ∨ q)) true? Call this number N.

(q ∧ (¬r ∨ ¬p)) → (¬p ∧ ¬(r ∨ q))
N =
5.Hard
∀x ∀y ¬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∀x ∀y ¬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
pqrp → qr → qp → r111110101100011010001000

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

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

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