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 ¬((¬p → (q ↔ r)) ∧ (p → ¬q)) have?

¬((¬p → (q ↔ r)) ∧ (p → ¬q))
N =
2.Easy
abb ↔ (b ∧ ¬a)11?10?01?00?

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

b ↔ (b ∧ ¬a)
N =
3.Medium
r ↔ ¬qr ∧ ¬q¬r ↔ ¬q¬r ∨ ¬q

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

¬(r ↔ q)
  1. 1r ↔ ¬q
  2. 2r ∧ ¬q
  3. 3¬r ↔ ¬q
  4. 4¬r ∨ ¬q
4.Medium
¬(p ∧ q) → ¬¬r¬(p ∧ q) ∧ ¬r¬(p ∧ q) ∧ ¬¬r(p ∧ q) ∧ ¬¬r

You want to prove (p ∧ q) → ¬r by contradiction. Which statement do you assume at the start? Click it.

(p ∧ q) → ¬r
  1. 1¬(p ∧ q) → ¬¬r
  2. 2¬(p ∧ q) ∧ ¬r
  3. 3¬(p ∧ q) ∧ ¬¬r
  4. 4(p ∧ q) ∧ ¬¬r
5.Hard
pqrq → pr → pq → r111110101100011010001000

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

q → pr → p∴ q → r
  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
6.Hard
∃x F(x)∀x F(x)∃x ¬F(x)∀x ¬F(x)

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

¬∃x F(x)
  1. 1∃x F(x)
  2. 2∀x F(x)
  3. 3∃x ¬F(x)
  4. 4∀x ¬F(x)

Answer key

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

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