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 (¬(¬b → a) → ¬c) ∨ (b ↔ c) have?

(¬(¬b → a) → ¬c) ∨ (b ↔ c)
N =
2.Easy
pq¬(¬(p ↔ q) → ¬q)11100100

Exactly one row of the truth table makes ¬(¬(p ↔ q) → ¬q) true. Click that row.

¬(¬(p ↔ q) → ¬q)
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
3.Medium
¬r ∧ ¬p¬p → ¬¬rp → ¬r¬¬r → ¬p

You want to prove ¬r → p by contrapositive. Which implication do you prove instead? Click it.

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

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

((q ∧ p) → r) → ((¬p ↔ r) ∧ ¬q)
N =
5.Hard

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

((¬r ∨ q) ∧ (s ∧ q)) ∨ ¬(s ↔ (¬p → 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 = 8
  2. 2. p = 0, q = 1
  3. 3. ¬p → ¬¬r
  4. 4. N = 3
  5. 5. N = 11
  6. 6. ∀x (F(x) ∧ ¬G(x))

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