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
aba → b11100100

Exactly one row of the truth table makes a → b false. Click that row.

a → b
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
2.Easy

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

((q ∧ ¬r) ↔ ¬q) ∨ ¬(p ∧ r)
N =
3.Medium
pqrq → pr ∨ (q → p)¬(¬p ∧ (¬r → q)) ∨ (r ∨ (q → p))111?110?101?100?011?010?001?000?

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

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

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

q ∧ (p ∨ r)
  1. 1q ∨ (p ∧ r)
  2. 2(q ∧ p) ∨ (q ∧ r)
  3. 3(q ∧ p) ∨ r
  4. 4(q ∨ p) ∧ (q ∨ r)
5.Hard
abb → aab11100100

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

b → aa∴ b
  1. 1a = 1, b = 1
  2. 2a = 1, b = 0
  3. 3a = 0, b = 1
  4. 4a = 0, b = 0
6.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)

Answer key

  1. 1. a = 1, b = 0
  2. 2. N = 2
  3. 3. N = 7
  4. 4. (q ∧ p) ∨ (q ∧ r)
  5. 5. a = 1, b = 0
  6. 6. ∀x ¬A(x)

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