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
pqq ↔ (q → p)11?10?01?00?

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

q ↔ (q → p)
N =
2.Easy
pqq → (p ∧ q)11100100

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

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
¬r ∧ ¬(q ∧ p)(q ∧ p) → ¬r¬¬r → ¬(q ∧ p)¬(q ∧ p) → ¬¬r

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

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

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

(p ↔ q) ↔ ¬(p ∨ (r ∨ q))
N =
5.Hard
∀x ¬A(x)∃x A(x)∃x ¬A(x)∀x A(x)

Which statement is equivalent to ¬∃x A(x), with the negation moved all the way inside? Click it.

¬∃x A(x)
  1. 1∀x ¬A(x)
  2. 2∃x A(x)
  3. 3∃x ¬A(x)
  4. 4∀x A(x)
6.Hard
prr → p¬r¬p11100100

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

r → p¬r∴ ¬p
  1. 1p = 1, r = 1
  2. 2p = 1, r = 0
  3. 3p = 0, r = 1
  4. 4p = 0, r = 0

Answer key

  1. 1. N = 1
  2. 2. p = 0, q = 1
  3. 3. ¬(q ∧ p) → ¬¬r
  4. 4. N = 5
  5. 5. ∀x ¬A(x)
  6. 6. p = 1, r = 0

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