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
abb ∨ ¬(b ↔ ¬a)11100100

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

b ∨ ¬(b ↔ ¬a)
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
2.Easy

How many rows N does the complete truth table of (c ↔ a) ∧ ((a → b) ∧ c) have?

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

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

(r ∨ ¬q) → ((p → q) ↔ r)
N =
4.Medium
pqr¬qp ↔ ¬q((q ∨ r) → r) → (p ↔ ¬q)111?110?101?100?011?010?001?000?

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

((q ∨ r) → r) → (p ↔ ¬q)
N =
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

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

((s ∧ q) ∧ (p ↔ r)) ∧ ((s ↔ ¬r) ∧ (p ↔ ¬q))
N =

Answer key

  1. 1. a = 1, b = 0
  2. 2. N = 8
  3. 3. N = 6
  4. 4. N = 5
  5. 5. ∃x (F(x) ∧ ∀y ¬R(x, y))
  6. 6. N = 1

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