INDIRECT PROOF
Definition Of Indirect Proof
Indirect proof is a type of proof in which a statement to be proved is assumed false and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true
Video Examples: Introduction to Indirect Proof
Example of Indirect Proof
Sum of 2n even numbers is even, where n > 0. Prove the statement using an indirect proof.
The first step of an indirect proof is to assume that 'Sum of even integers is odd.'
That is, 2 + 4 + 6 + 8 + . . . .+ 2n = an odd number
⇒ 2(1 + 2 + 3 + 4 + . . . + n) = an odd number ⇒ 2 × 
= an odd number
⇒ n(n + 1) = an odd number, a contradiction, because n(n + 1) is always an even number.
Thus, the statement is proved using an indirect proof.
Solved Example on Indirect Proof
Ques: Prove the following statement using an indirect proof: â–³LMN has at most one right angle.
Solution:
Step 1: Assume â–³LMN has more than one right angle. That is, assume that angle L and angle M are both right angles.
Step 2: If M and N are both right angles, then m∠L = m∠M = 90
Step 3: m∠L + m∠M + m∠N = 180 [The sum of the measures of the angles of a triangle is 180.]
Step 4: Substitution gives 90 + 90 + m∠N = 180.
Step 5: Solving gives m∠N = 0.
Step 6: This means that there is no â–³LMN, which contradicts the given statement.
Step 7: So, the assumption that ∠L and ∠M are both right angles must be false.
Step 8: Therefore, â–³LMN has at most one right angle.
Quick Summary
- Assume the opposite of what you want to prove.
- Show that this assumption leads to a contradiction.
- Conclude that the original statement must be true.
🍎 Teacher Insights
Emphasize the importance of clear and precise negation of the statement. Provide ample examples of both successful and unsuccessful attempts at indirect proofs. Encourage students to identify the specific contradiction that arises.🎓 Prerequisites
- Direct Proof
- Basic Algebra
- Understanding of Contradiction
Check Your Knowledge
Q1: Which of the following is the first step in an indirect proof?
Q2: What does it mean when an assumption in an indirect proof leads to a contradiction?
Frequently Asked Questions
Q: When should I use indirect proof?
A: Indirect proof is useful when a direct proof is difficult or impossible. It's often helpful when proving negative statements (e.g., 'not', 'no').
Q: What is a contradiction?
A: A contradiction is a situation where two statements are mutually exclusive and cannot both be true at the same time.