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

How many rows N does the complete truth table of ((p ∧ r) ↔ q) ∧ (p ∧ (¬q ∨ r)) have?

((p ∧ r) ↔ q) ∧ (p ∧ (¬q ∨ r))
N =
2.Easy
abc(c ∧ b) ∧ (c ↔ a)111?110?101?100?011?010?001?000?

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

(c ∧ b) ∧ (c ↔ a)
N =
3.Medium
¬a → ¬ca ∧ ¬c¬a ∧ c¬a ∨ ¬c

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

¬(a → c)
  1. 1¬a → ¬c
  2. 2a ∧ ¬c
  3. 3¬a ∧ c
  4. 4¬a ∨ ¬c
4.Medium
(q → r) ∨ (r → q)(q → r) ∧ (¬r → ¬q)q ∧ r(q → r) ∧ (r → q)

Which formula is logically equivalent to q ↔ r? Click it.

q ↔ r
  1. 1(q → r) ∨ (r → q)
  2. 2(q → r) ∧ (¬r → ¬q)
  3. 3q ∧ r
  4. 4(q → r) ∧ (r → 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 ∃y ¬R(x, y)∃y ∀x ¬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∃y ∀x ¬R(x, y)
  3. 3∃x ∀y ¬R(x, y)
  4. 4∀x ∀y ¬R(x, y)

Answer key

  1. 1. N = 8
  2. 2. N = 1
  3. 3. a ∧ ¬c
  4. 4. (q → r) ∧ (r → q)
  5. 5. ∃x (F(x) ∧ ∀y ¬R(x, y))
  6. 6. ∃x ∀y ¬R(x, y)

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