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) ↔ q11100100

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

(p ∨ q) ↔ q
  1. 1p = 1, q = 1
  2. 2p = 1, q = 0
  3. 3p = 0, q = 1
  4. 4p = 0, q = 0
2.Easy

How many rows N does the complete truth table of (¬(b ∧ c) ∨ a) ∧ (b ∧ ¬c) have?

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

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

(p ∨ q) → r
  1. 1(p → r) ∧ (q → r)
  2. 2p ∨ (q → r)
  3. 3(p ∧ q) → r
  4. 4(p → r) ∨ (q → r)
4.Medium
¬c ∨ ¬a¬c ∧ a¬c → ¬ac ∧ ¬a

Which formula is logically equivalent to ¬(c → a)? Click it.

¬(c → a)
  1. 1¬c ∨ ¬a
  2. 2¬c ∧ a
  3. 3¬c → ¬a
  4. 4c ∧ ¬a
5.Hard

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

((r ∨ p) → s) → ((¬r → q) ∧ s)
N =
6.Hard
∀x (A(x) ∧ ¬B(x))∀x (A(x) → ¬B(x))∃x (A(x) ∧ ¬B(x))∃x (¬A(x) ∧ B(x))

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

¬∀x (A(x) → B(x))
  1. 1∀x (A(x) ∧ ¬B(x))
  2. 2∀x (A(x) → ¬B(x))
  3. 3∃x (A(x) ∧ ¬B(x))
  4. 4∃x (¬A(x) ∧ B(x))

Answer key

  1. 1. p = 1, q = 0
  2. 2. N = 8
  3. 3. (p → r) ∧ (q → r)
  4. 4. c ∧ ¬a
  5. 5. N = 12
  6. 6. ∃x (A(x) ∧ ¬B(x))

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