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) ∧ b)11?10?01?00?

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

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

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

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
b ∧ (c ∨ a)(b ∧ c) ∨ (b ∧ a)(b ∨ c) ∧ a(b ∨ c) ∧ (b ∨ a)

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

b ∨ (c ∧ a)
  1. 1b ∧ (c ∨ a)
  2. 2(b ∧ c) ∨ (b ∧ a)
  3. 3(b ∨ c) ∧ a
  4. 4(b ∨ c) ∧ (b ∨ a)
4.Medium
c ∧ (b → a)c → (b → a)(c → a) ∧ (b → a)(c → b) → a

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

(c ∧ b) → a
  1. 1c ∧ (b → a)
  2. 2c → (b → a)
  3. 3(c → a) ∧ (b → a)
  4. 4(c → b) → a
5.Hard
∃x (P(x) → ∀y ¬R(x, y))∃x (P(x) ∧ ∀y ¬R(x, y))∀x (P(x) ∧ ∀y ¬R(x, y))∃x (¬P(x) ∧ ∃y R(x, y))

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

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

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

¬(¬s ↔ ¬q) ∨ ((p ∨ r) → (r ∨ s))
N =

Answer key

  1. 1. N = 3
  2. 2. a = 1, b = 1
  3. 3. (b ∨ c) ∧ (b ∨ a)
  4. 4. c → (b → a)
  5. 5. ∃x (P(x) ∧ ∀y ¬R(x, y))
  6. 6. N = 15

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