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Indirect Proof

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Indirect Proof

Study Guide

Key Definition

An indirect proof is a method of proving a statement by assuming the negation of the desired conclusion and showing that this assumption leads to a contradiction.

Important Notes

Indirect proof involves assuming the opposite of what you want to prove.
Proof by contradiction is a common form of indirect proof.
$\neg q$ leads to a logical contradiction, proving $q$ must be true.
Indirect proof can be used to prove theorems in various branches of mathematics.
Always verify that the contradiction is genuine and not due to an error in logic.

Mathematical Notation

$\neg$ represents negation

$\therefore$ means 'therefore'

$\because$ means 'because'

$\Rightarrow$ indicates implication

$\Leftrightarrow$ indicates equivalence

Use ¬ to denote negation and ⇒ for implication consistently.

Why It Works

By assuming the negation of the statement and deriving a contradiction with known premises or axioms, we confirm the original statement must hold.

Remember

Link your assumed negation back to a specific fact or theorem to reveal a genuine contradiction.

Quick Reference

Property:$\neg p \Rightarrow \text{Contradiction}$

Understanding Indirect Proof

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Video explanation of this concept

Beginner

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Beginner Explanation

In an indirect proof, you assume the opposite of what you want to prove. If this assumption leads to a contradiction, the original statement is true.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which statement is true for an indirect proof? Assume $\neg q$ leads to a contradiction.

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Prove that if a person is fluent in English, they can attend an English-speaking school. Assume $p$ is 'fluent in English' and $q$ is 'can attend an English-speaking school'.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the statement: 'If it rains, the ground is wet.' Assume $\neg q$ leads to a contradiction.

Challenge Quiz

Single Choice Quiz

Advanced

Select the correct conclusion for: 'If $x^2 ≥ 0$ for all $x$, then $x$ is real.' Assume $¬q$ leads to a contradiction.

Recap

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Review key concepts and takeaways