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
pq(¬p → q) → (¬p → ¬q)11?10?01?00?

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

(¬p → q) → (¬p → ¬q)
N =
2.Easy
pqr((q ∧ r) ∧ p) ↔ (p ∨ ¬q)111?110?101?100?011?010?001?000?

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

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

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

((¬b ∨ ¬c) ∧ (a ↔ ¬c)) → (b ∨ ¬a)
N =
4.Medium
¬r ∧ q¬r ∧ ¬qr ∨ ¬q¬r ∨ ¬q

Which formula is logically equivalent to ¬(r ∨ q)? Click it.

¬(r ∨ q)
  1. 1¬r ∧ q
  2. 2¬r ∧ ¬q
  3. 3r ∨ ¬q
  4. 4¬r ∨ ¬q
5.Hard
∃x (¬P(x) ∨ ¬Q(x))∀x (¬P(x) ∧ ¬Q(x))∃x (¬P(x) ∧ ¬Q(x))∀x (¬P(x) ∨ ¬Q(x))

Which statement is equivalent to ¬∀x (P(x) ∨ Q(x)), with the negation moved all the way inside? Click it.

¬∀x (P(x) ∨ Q(x))
  1. 1∃x (¬P(x) ∨ ¬Q(x))
  2. 2∀x (¬P(x) ∧ ¬Q(x))
  3. 3∃x (¬P(x) ∧ ¬Q(x))
  4. 4∀x (¬P(x) ∨ ¬Q(x))
6.Hard

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

¬(q ∨ (p ↔ r)) ∨ (s ∧ ¬(p → ¬r))
N =

Answer key

  1. 1. N = 3
  2. 2. N = 3
  3. 3. N = 7
  4. 4. ¬r ∧ ¬q
  5. 5. ∃x (¬P(x) ∧ ¬Q(x))
  6. 6. N = 6

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