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 Indirect Proof= 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: ?LMN has at most one right angle.

Solution:

    Step 1: Assume ?LMN 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 ?LMN, which contradicts the given statement.
    Step 7: So, the assumption that ?L and ?M are both right angles must be false.
    Step 8: Therefore, ?LMN has at most one right angle.