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(¬b → a) ∨ (a ∧ b)11?10?01?00?

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

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

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

¬((¬b ∨ a) ∧ ¬(b ∨ a))
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
3.Medium
r ∨ (q ∧ p)(r ∧ q) ∨ p(r ∧ q) ∨ (r ∧ p)(r ∨ q) ∧ (r ∨ p)

Which formula is logically equivalent to r ∧ (q ∨ p)? Click it.

r ∧ (q ∨ p)
  1. 1r ∨ (q ∧ p)
  2. 2(r ∧ q) ∨ p
  3. 3(r ∧ q) ∨ (r ∧ p)
  4. 4(r ∨ q) ∧ (r ∨ p)
4.Medium
pqrq ∨ p¬(q ∨ p)((r → p) ∧ q) → ¬(q ∨ p)111?110?101?100?011?010?001?000?

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

((r → p) ∧ q) → ¬(q ∨ p)
N =
5.Hard

How many of the 16 rows of the truth table make (¬q ∨ (¬q → p)) ↔ (¬s ∧ r) true? Call this number N.

(¬q ∨ (¬q → p)) ↔ (¬s ∧ r)
N =
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. N = 3
  2. 2. a = 0, b = 0
  3. 3. (r ∧ q) ∨ (r ∧ p)
  4. 4. N = 5
  5. 5. N = 4
  6. 6. ∀x (F(x) ∧ ¬G(x))

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