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What is a Counter-Example?
Grade Level:
Class 2
Logic, Maths, AI, Law, Science
Definition
What is it?
A counter-example is a specific example that proves a general statement or idea is false. If someone says 'all mangoes are yellow,' you can show them a green mango to prove they are wrong. That green mango is a counter-example.
Simple Example
Quick Example
Imagine your friend says, 'All students in our school wear blue uniforms.' You know that Class 10 students wear white shirts on Fridays for special events. So, a Class 10 student wearing a white shirt on Friday is a counter-example to your friend's statement.
Worked Example
Step-by-Step
Statement: 'All birds can fly.'
---Step 1: Understand the statement. It says EVERY single bird has the ability to fly.
---Step 2: Think of different types of birds. Sparrow, crow, parrot, eagle.
---Step 3: Can you think of any bird that cannot fly? Yes, a penguin or an ostrich!
---Step 4: Pick one. A penguin cannot fly.
---Step 5: So, a penguin is a counter-example to the statement 'All birds can fly.'
Answer: A penguin is a counter-example.
Why It Matters
Counter-examples are super important in maths, science, and even daily life to check if ideas are truly correct. Scientists use them to test theories, and lawyers use them to find flaws in arguments. Learning this helps you think critically and solve problems like a real detective!
Common Mistakes
MISTAKE: Giving an example that fits the statement instead of disproving it. | CORRECTION: A counter-example MUST show the statement is wrong, not right.
MISTAKE: Thinking one wrong example isn't enough to disprove a general statement. | CORRECTION: Even ONE single counter-example is enough to prove a general statement false.
MISTAKE: Confusing a counter-example with an exception that still fits a broader rule. | CORRECTION: A counter-example directly breaks the rule being stated, not just an unusual case within it.
Practice Questions
Try It Yourself
QUESTION: Your sister says, 'All Indian food is spicy.' Give a counter-example. | ANSWER: Kheer (a sweet rice pudding) or plain curd rice are counter-examples.
QUESTION: Statement: 'All even numbers are divisible by 4.' Is this true or false? If false, give a counter-example. | ANSWER: False. A counter-example is 6 (it's even but not divisible by 4).
QUESTION: Your friend claims, 'Every person who studies hard gets 100% in all exams.' Think about real life and give a counter-example to this statement. Explain why it disproves the claim. | ANSWER: A student who studies very hard but gets 95% in an exam is a counter-example. This disproves the claim because even with hard study, they did not get 100% in ALL exams.
MCQ
Quick Quiz
Which of the following is the best counter-example to the statement: 'All fruits are red'?
An apple
A strawberry
A banana
A cherry
The Correct Answer Is:
C
A banana is yellow, which proves that not all fruits are red. Apples, strawberries, and cherries can all be red, so they don't disprove the statement.
Real World Connection
In the Real World
When a software engineer develops a new app, they often make general statements about how it will work. Testers then try to find 'bugs' or counter-examples – specific ways the app doesn't work as expected. For instance, if an app is supposed to accept all Indian mobile numbers, a tester might try entering a number with 9 digits instead of 10. If it crashes, that's a counter-example!
Key Vocabulary
Key Terms
STATEMENT: A claim or idea that needs to be checked. | PROOF: Evidence that shows something is true or false. | FALSE: Not true or incorrect. | GENERALIZATION: A broad statement that applies to many things.
What's Next
What to Learn Next
Now that you understand counter-examples, you can move on to learning about 'Logical Reasoning' and 'Deductive vs. Inductive Arguments'. These concepts will help you build strong arguments and identify weak ones, using what you've learned about proving statements false.
