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Counterexample Generation Beginner-Intermediate Worksheet: Focus on common variations practice Counterexample Generation BEGINNER INTERMEDIATE

Level up your Counterexample Generation skills! You're at Worksheet 4 of 10 (33% through this series). This step-up challenge worksheet features 20 beginner-intermediate-level problems with a focus on common variations practice. Topics covered: counterexample generation for competitive exams, how to solve counterexample generation, counterexample generation tricks.

📝 Worksheet 4 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Beginner Intermediate level

What you'll learn in this worksheet:
Your progress through Counterexample Generation
Worksheet 4 of 10 (33% complete)

Question 1

Find a counterexample to show this statement is FALSE: "p ∧ (q ∨ r) is equivalent to (p ∧ q) ∨ r" Provide truth values for p, q, r that make the two sides different.
∧ doesn't distribute over ∨ that way. Correct: p∧(q∨r) ≡ (p∧q) ∨ (p∧r)

p=False, q=True, r=False: Left=False, Right=True → Different!

Question 2

Find a counterexample to show this statement is FALSE: "p → q is logically equivalent to q → p" Provide truth values for p, q, r that make the two sides different.
Implication is not symmetric. Converse is not equivalent to original.

p=True, q=False: p→q=False, q→p=True → Different!

Question 3

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 4

Find a counterexample to show this statement is FALSE: "p ∨ q is logically equivalent to p ⊕ q" Provide truth values for p, q, r that make the two sides different.
Inclusive OR is true when both are true; exclusive OR is false when both are true.

p=True, q=True: p∨q=True, p⊕q=False → Different!

Question 5

Find a counterexample to show this statement is FALSE: "p ∨ q is logically equivalent to p ⊕ q" Provide truth values for p, q, r that make the two sides different.
Inclusive OR is true when both are true; exclusive OR is false when both are true.

p=True, q=True: p∨q=True, p⊕q=False → Different!

Question 6

Find a counterexample to show this statement is FALSE: "p ∨ q is logically equivalent to p ⊕ q" Provide truth values for p, q, r that make the two sides different.
Inclusive OR is true when both are true; exclusive OR is false when both are true.

p=True, q=True: p∨q=True, p⊕q=False → Different!

Question 7

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 8

Find a counterexample to show this statement is FALSE: "p ∨ q is logically equivalent to p ⊕ q" Provide truth values for p, q, r that make the two sides different.
Inclusive OR is true when both are true; exclusive OR is false when both are true.

p=True, q=True: p∨q=True, p⊕q=False → Different!

Question 9

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 10

Find a counterexample to show this statement is FALSE: "p ∨ q is logically equivalent to p ⊕ q" Provide truth values for p, q, r that make the two sides different.
Inclusive OR is true when both are true; exclusive OR is false when both are true.

p=True, q=True: p∨q=True, p⊕q=False → Different!

Question 11

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 12

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 13

Find a counterexample to show this statement is FALSE: "p ∧ (q ∨ r) is equivalent to (p ∧ q) ∨ r" Provide truth values for p, q, r that make the two sides different.
∧ doesn't distribute over ∨ that way. Correct: p∧(q∨r) ≡ (p∧q) ∨ (p∧r)

p=False, q=True, r=False: Left=False, Right=True → Different!

Question 14

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 15

Find a counterexample to show this statement is FALSE: "p ∧ (q ∨ r) is equivalent to (p ∧ q) ∨ r" Provide truth values for p, q, r that make the two sides different.
∧ doesn't distribute over ∨ that way. Correct: p∧(q∨r) ≡ (p∧q) ∨ (p∧r)

p=False, q=True, r=False: Left=False, Right=True → Different!

Question 16

Find a counterexample to show this statement is FALSE: "p → q is logically equivalent to q → p" Provide truth values for p, q, r that make the two sides different.
Implication is not symmetric. Converse is not equivalent to original.

p=True, q=False: p→q=False, q→p=True → Different!

Question 17

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 18

Find a counterexample to show this statement is FALSE: "p ∧ (q ∨ r) is equivalent to (p ∧ q) ∨ r" Provide truth values for p, q, r that make the two sides different.
∧ doesn't distribute over ∨ that way. Correct: p∧(q∨r) ≡ (p∧q) ∨ (p∧r)

p=False, q=True, r=False: Left=False, Right=True → Different!

Question 19

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Question 20

Find a counterexample to show this statement is FALSE: "p → q is equivalent to ¬p → ¬q" Provide truth values for p, q, r that make the two sides different.
Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!
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