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(q ∧ p) ↔ (p → q)11?10?01?00?

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

(q ∧ p) ↔ (p → q)
N =
2.Easy
ab¬(¬b ∨ (¬a ↔ b))11100100

Exactly one row of the truth table makes ¬(¬b ∨ (¬a ↔ b)) true. Click that row.

¬(¬b ∨ (¬a ↔ b))
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
3.Medium
abcb → (c ∨ ¬a)¬(b → (c ∨ ¬a))(c → b) → ¬(b → (c ∨ ¬a))111?110?101?100?011?010?001?000?

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

(c → b) → ¬(b → (c ∨ ¬a))
N =
4.Medium
¬r ∧ q¬r ∧ ¬q¬r → ¬qr ∧ ¬q

You want to prove r → q by contradiction. Which statement do you assume at the start? Click it.

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

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

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

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

¬∀x F(x)
  1. 1∀x F(x)
  2. 2∃x ¬F(x)
  3. 3∃x F(x)
  4. 4∀x ¬F(x)

Answer key

  1. 1. N = 2
  2. 2. a = 1, b = 1
  3. 3. N = 3
  4. 4. r ∧ ¬q
  5. 5. ∃x (F(x) ∧ ∀y ¬R(x, y))
  6. 6. ∃x ¬F(x)

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