What's the best way to master Counterexample Generation quickly?

Master Counterexample Generation by understanding the core concepts first, practicing regularly with our worksheets, and applying the shortcut techniques provided.

Counterexample Generation

Counterexample Generation problems ask you to find a truth assignment to variables that makes a given logical statement false. This disproves the statement's claim of logical truth (tautology) or equivalence. These problems test your ability to find falsifying assignments.

10Worksheets

200+Practice Questions

MediumDifficulty

2-3 hoursHours to Master

What You'll Learn

Introduction & Overview Why This Matters How to Solve Pro Tips & Tricks

Shortcut Methods Common Mistakes Practice Worksheets Exam Importance

Introduction to Counterexample Generation

Prerequisites

Truth table concepts Logical equivalence understanding Finding falsifying assignments Counterexample construction

Why This Matters: Counterexample Generation appears in 1-2 questions in SSC CGL and Banking PO exams. It tests logical falsification skills.

How to Solve Counterexample Generation Problems

Step 1: Identify the claim being made (usually an equivalence or tautology)

Step 2: To disprove an equivalence, find inputs where outputs differ

Step 3: To disprove a tautology, find inputs where expression is false

Step 4: Work backwards from the desired false output

Step 5: Choose truth values that make the expression false

Step 6: Verify that the chosen values indeed make the expression false

Step 7: Present the counterexample truth assignment

Pro Strategy: To disprove a statement, find just one case where it fails. For equivalences, look for cases where the two expressions differ. For tautologies, look for any false case.

Example Problem

Example: Find a counterexample to show p → q is NOT equivalent to q → p. Solution: Step 1: Claim: p→q ≡ q→p Step 2: Find inputs where truth values differ Step 3: Try p=T, q=F: p→q = F, q→p = T → different Step 4: Counterexample: p=True, q=False Answer: p=True, q=False

Pro Tips & Tricks

For p→q, the only false case is T→F
For p↔q, false cases are T,F and F,T
For p∧q, false when at least one is false
For p∨q, false only when both false
For ¬p, false when p is true
To disprove an equivalence, find a row where truth values differ

Shortcut Methods to Solve Faster

To disprove p→q ≡ something, try p=T,q=F

To disprove p↔q ≡ something, try p=T,q=F or p=F,q=T

To disprove a tautology, try to make it false

Work backwards: decide what output you want, then solve for inputs

Common Mistakes to Avoid

Trying to prove rather than disprove

Finding a case where the statement is true (doesn't disprove)

Not verifying the counterexample

Assuming one counterexample doesn't exist when it does

Practice Worksheets

Practice makes perfect! Work through these worksheets to master Counterexample Generation. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.

Worksheet 1 Beginner Beginner level - Perfect for building foundational understanding

Worksheet 2 Beginner Beginner level - Perfect for building foundational understanding

Worksheet 3 Beginner Beginner level - Perfect for building foundational understanding

Worksheet 4 Intermediate Intermediate level - Test your understanding with moderate difficulty

Worksheet 5 Intermediate Intermediate level - Test your understanding with moderate difficulty

Worksheet 6 Intermediate Intermediate level - Test your understanding with moderate difficulty

Worksheet 7 Advanced Advanced level - Challenge yourself with complex problems

Worksheet 8 Advanced Advanced level - Challenge yourself with complex problems

Worksheet 9 Advanced Advanced level - Challenge yourself with complex problems

Worksheet 10 Advanced Advanced level - Challenge yourself with complex problems