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)11100100

Exactly one row of the truth table makes (p ∨ ¬q) ∧ (¬p → ¬q) false. 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
2.Easy

The complete truth table of ¬(r → q) ∧ (r → p) is written out. In how many rows N is q = 1 and r = 0?

¬(r → q) ∧ (r → p)
N =
3.Medium
b ∨ ¬c¬b ∨ c¬b → ¬cc → b

Which formula is logically equivalent to b → c? Click it.

b → c
  1. 1b ∨ ¬c
  2. 2¬b ∨ c
  3. 3¬b → ¬c
  4. 4c → b
4.Medium
¬a ∨ ¬b¬a ∨ b¬a ∧ b¬a ∧ ¬b

Which formula is logically equivalent to ¬(a ∧ b)? Click it.

¬(a ∧ b)
  1. 1¬a ∨ ¬b
  2. 2¬a ∨ b
  3. 3¬a ∧ b
  4. 4¬a ∧ ¬b
5.Hard
∃x ∃y (R(x, y) ∧ ¬R(y, x))∃x ∃y (R(x, y) → ¬R(y, x))∃x ∃y (¬R(x, y) ∧ ¬R(y, x))∀x ∀y (R(x, y) ∧ ¬R(y, x))

Which statement is equivalent to ¬∀x ∀y (R(x, y) → R(y, x)), with the negation moved all the way inside? Click it.

¬∀x ∀y (R(x, y) → R(y, x))
  1. 1∃x ∃y (R(x, y) ∧ ¬R(y, x))
  2. 2∃x ∃y (R(x, y) → ¬R(y, x))
  3. 3∃x ∃y (¬R(x, y) ∧ ¬R(y, x))
  4. 4∀x ∀y (R(x, y) ∧ ¬R(y, x))
6.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

Answer key

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

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