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
abc¬(¬(c ∨ ¬a) → (b → (¬b ∨ c)))111?110?101?100?011?010?001?000?

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

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

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

¬((p ∧ q) → (p → ¬q))
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
3.Medium
pqrq ∧ (p ∨ q)¬(q ∧ (p ∨ q))((¬p ∧ ¬r) ∨ ¬r) ∨ ¬(q ∧ (p ∨ q))111?110?101?100?011?010?001?000?

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

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

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

(r ∨ q) ∨ ¬((r ↔ p) ↔ q)
N =
5.Hard

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

((p ∨ q) ↔ (r ↔ q)) ∧ ¬(r → s)
N =
6.Hard
∃x ∀y ¬R(x, y)∃x ∃y ¬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∃x ∃y ¬R(x, y)
  3. 3∀x ∃y ¬R(x, y)
  4. 4∀x ∀y ¬R(x, y)

Answer key

  1. 1. N = 1
  2. 2. p = 1, q = 1
  3. 3. N = 6
  4. 4. N = 7
  5. 5. N = 3
  6. 6. ∀x ∃y ¬R(x, y)

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