What is Proof by Contradiction?

S3-SA1-0407

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Proof by Contradiction is a clever way to prove something is true by first assuming it is false. If this assumption leads to a situation that cannot be true (a contradiction), then our original assumption must be wrong. This means the original statement we wanted to prove must be true.

Simple Example

Quick Example

Imagine your friend says, 'I scored 100% in the Maths exam!' You want to prove this is true. Using contradiction, you first assume your friend did NOT score 100%. If you then find out their report card clearly shows 'Maths: 100%', this is a contradiction. So, your initial assumption (they didn't score 100%) must be false, meaning they actually did score 100%.

Worked Example

Step-by-Step

Let's prove that it's impossible for a triangle to have two right angles (90 degrees each).

1. ASSUME THE OPPOSITE: Let's assume a triangle CAN have two right angles. So, Triangle ABC has Angle A = 90 degrees and Angle B = 90 degrees.
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2. RECALL A KNOWN FACT: We know that the sum of angles in any triangle is always 180 degrees (Angle A + Angle B + Angle C = 180 degrees).
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3. APPLY THE ASSUMPTION: If Angle A = 90 degrees and Angle B = 90 degrees, then 90 + 90 + Angle C = 180 degrees.
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4. SIMPLIFY: This means 180 + Angle C = 180 degrees.
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5. FIND THE CONTRADICTION: Subtracting 180 from both sides, we get Angle C = 0 degrees. But an angle in a triangle cannot be 0 degrees; it must be greater than 0.
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6. CONCLUDE: This result (Angle C = 0 degrees) is a contradiction! It means our initial assumption (that a triangle can have two right angles) must be false.
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7. FINAL PROOF: Therefore, it is impossible for a triangle to have two right angles. The original statement is true.

Why It Matters

This powerful proof technique is used in various advanced fields to establish fundamental truths. Scientists use it in Physics and Computer Science to prove theories or verify algorithms. It helps engineers design reliable systems and is crucial in Cryptography to ensure data security, making our online transactions safe.

Common Mistakes

MISTAKE: Assuming the opposite of the statement is the same as just proving the original statement wrong. | CORRECTION: The 'opposite' (or negation) must be a clear, logical reversal of the original statement. For example, if the statement is 'All mangoes are sweet', the opposite is 'At least one mango is not sweet', not 'All mangoes are sour'.

MISTAKE: Failing to find a genuine contradiction, or creating a contradiction based on incorrect logic. | CORRECTION: The contradiction must arise from a known, undeniable truth or a logical impossibility. Double-check every step to ensure the reasoning is sound and doesn't introduce new errors.

MISTAKE: Stopping after finding the contradiction without clearly stating why it proves the original statement. | CORRECTION: After finding the contradiction, explicitly state that because the assumption led to an impossible situation, the assumption must be false, and therefore the original statement must be true.

Practice Questions

Try It Yourself

QUESTION: A shopkeeper claims he has 100 kg of rice. If you assume he does NOT have 100 kg, and then you weigh the rice and it comes out to exactly 100 kg, what does this contradiction prove? | ANSWER: It proves that the shopkeeper's original claim (he has 100 kg of rice) is true.

QUESTION: Prove by contradiction: There is no largest even number. (Hint: Assume there IS a largest even number, say 'N'. What happens if you add 2 to N?) | ANSWER: Assume there IS a largest even number, N. Then N+2 would also be an even number, and N+2 is clearly larger than N. This contradicts our assumption that N was the largest even number. Therefore, our assumption is false, and there is no largest even number.

QUESTION: Suppose your school's annual sports day has only two events: a 100-meter race and a long jump. Every student must participate in at least one event. Can there be a student who participates in NEITHER event? Use proof by contradiction. | ANSWER: Assume there IS a student who participates in NEITHER event. The rule states 'Every student must participate in AT LEAST ONE event'. Our assumption (participates in neither) directly contradicts this rule. Since our assumption leads to a contradiction, the assumption must be false. Therefore, there cannot be a student who participates in neither event.

MCQ

Quick Quiz

What is the first step in Proof by Contradiction?

State the conclusion you want to prove.

Assume the opposite of what you want to prove.

Look for an example that supports the statement.

Directly prove the statement is true.

The Correct Answer Is:

The core idea of Proof by Contradiction is to start by assuming the statement you want to prove is false (or its opposite is true). Options A, C, and D describe other parts of proving or different proof methods.

Real World Connection

In the Real World

Imagine an app like Google Maps or Ola Cabs. When it calculates the shortest route, it might use a form of contradiction. If it assumes a certain path is NOT the shortest, but then finds that any other path is longer, it contradicts the initial assumption, confirming the first path IS the shortest. This helps you reach your destination faster, whether by auto or car!

Key Vocabulary

Key Terms

ASSUMPTION: A statement taken to be true for the sake of argument | CONTRADICTION: A situation where two statements cannot both be true at the same time; a logical impossibility | PROOF: A logical argument that establishes the truth of a statement | NEGATION: The opposite of a statement

What's Next

What to Learn Next

Now that you understand Proof by Contradiction, you can explore other proof methods like Direct Proof or Proof by Induction. These techniques are fundamental in higher mathematics and will help you solve even more complex problems in your future studies.