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
ab(b ∨ ¬a) ∨ ¬(b → a)11?10?01?00?

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

(b ∨ ¬a) ∨ ¬(b → a)
N =
2.Easy

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

((¬r ∧ p) ∧ q) ↔ ((p ∨ q) ∨ r)
N =
3.Medium
¬p → ¬qp → q¬p ∧ ¬q¬q → ¬p

Which formula is logically equivalent to q → p? Click it.

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

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

(q ∧ (¬r ↔ p)) ∧ (p ↔ q)
N =
5.Hard
∃x ∃y ¬R(x, y)∀x ∃y ¬R(x, y)∀x ∀y ¬R(x, y)∃x ∀y ¬R(x, y)

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

¬∃x ∀y R(x, y)
  1. 1∃x ∃y ¬R(x, y)
  2. 2∀x ∃y ¬R(x, y)
  3. 3∀x ∀y ¬R(x, y)
  4. 4∃x ∀y ¬R(x, y)
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 = 3
  2. 2. N = 2
  3. 3. ¬p → ¬q
  4. 4. N = 1
  5. 5. ∀x ∃y ¬R(x, y)
  6. 6. ∃x ¬F(x)

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