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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
pq(p ∨ q) ∧ (¬q ∨ ¬p)11?10?01?00?

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

(p ∨ q) ∧ (¬q ∨ ¬p)
N =
2.Easy
pqr¬(q → r) ∨ ((p ∧ r) → q)111110101100011010001000

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

¬(q → r) ∨ ((p ∧ r) → q)
  1. 1p = 1, q = 1, r = 1
  2. 2p = 1, q = 1, r = 0
  3. 3p = 1, q = 0, r = 1
  4. 4p = 1, q = 0, r = 0
  5. 5p = 0, q = 1, r = 1
  6. 6p = 0, q = 1, r = 0
  7. 7p = 0, q = 0, r = 1
  8. 8p = 0, q = 0, r = 0
3.Medium
pqrr → q(r → q) → p¬(r ∧ p) ∧ ((r → q) → p)111?110?101?100?011?010?001?000?

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

¬(r ∧ p) ∧ ((r → q) → p)
N =
4.Medium
(r → q) ∧ (p → q)(r → q) ∨ (p → q)r ∨ (p → q)(r ∧ p) → q

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

(r ∨ p) → q
  1. 1(r → q) ∧ (p → q)
  2. 2(r → q) ∨ (p → q)
  3. 3r ∨ (p → q)
  4. 4(r ∧ p) → q
5.Hard
bcb → ccb11100100

The argument with premises b → c; c and conclusion b is invalid. Click the row of the truth table that is a counterexample.

b → cc∴ b
  1. 1b = 1, c = 1
  2. 2b = 1, c = 0
  3. 3b = 0, c = 1
  4. 4b = 0, c = 0
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 = 2
  2. 2. p = 1, q = 0, r = 1
  3. 3. N = 3
  4. 4. (r → q) ∧ (p → q)
  5. 5. b = 0, c = 1
  6. 6. ∀x ∃y ¬R(x, y)

Full step-by-step solutions: practise this topic on examproblems.com.