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 → 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
2.Easy
pqq → ¬(p ∨ q)11?10?01?00?

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

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

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

((¬a ∨ c) ↔ b) → (a → c)
N =
4.Medium
a → c¬c → ¬ac ∨ ¬a¬c ∨ a

Which formula is logically equivalent to c → a? Click it.

c → a
  1. 1a → c
  2. 2¬c → ¬a
  3. 3c ∨ ¬a
  4. 4¬c ∨ a
5.Hard
∃x (A(x) → ¬B(x))∃x (A(x) ∧ ¬B(x))∀x (A(x) → ¬B(x))∃x (¬A(x) ∧ B(x))

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

¬∀x (A(x) → B(x))
  1. 1∃x (A(x) → ¬B(x))
  2. 2∃x (A(x) ∧ ¬B(x))
  3. 3∀x (A(x) → ¬B(x))
  4. 4∃x (¬A(x) ∧ B(x))
6.Hard
∀x (A(x) → ¬B(x))∀x (¬A(x) ∧ ¬B(x))∃x (¬A(x) ∧ ¬B(x))∀x (A(x) ∧ ¬B(x))

Which statement is equivalent to ¬∃x (A(x) ∧ B(x)), with the negation moved all the way inside? Click it.

¬∃x (A(x) ∧ B(x))
  1. 1∀x (A(x) → ¬B(x))
  2. 2∀x (¬A(x) ∧ ¬B(x))
  3. 3∃x (¬A(x) ∧ ¬B(x))
  4. 4∀x (A(x) ∧ ¬B(x))

Answer key

  1. 1. a = 0, b = 1
  2. 2. N = 2
  3. 3. N = 7
  4. 4. ¬c ∨ a
  5. 5. ∃x (A(x) ∧ ¬B(x))
  6. 6. ∀x (A(x) → ¬B(x))

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