1.3.2

Proof by Deduction

Test yourself

What is proof by deduction?

Proof by deduction is a proof that consists of using known theorems to prove a given statement is always true.

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How do you prove by deduction?

  • Proof by deduction begins with a known theorem.
  • We use a series of logical steps to go from the theorem to the final statement.
    • You need to write out each step clearly and consider all potential cases.
  • This statement should either prove the conjecture is always true or that it must be false.
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Example

  • Prove that the sum of two odd numbers is even.
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Assumptions

  • If xx and yy are odd, we can write them as:
    • x=2a+1x = 2a+1 and y=2b+1y = 2b+1
  • Where aa and bb are integers.
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Sum

  • Adding xx and yy together, we get:
    • x+y=2a+1+2b+1=2a+2b+2x+y = 2a+1+2b+1 = 2a + 2b + 2
  • Collecting terms in a sum like this is known to be always true, so it's a valid step in the proof.
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Inspection

  • In order for x+yx+y to be even, we need to be able to write it as 2c2c, where cc is an integer.
  • By inspecting the equation for x+yx+y we obtained earlier, we can rewrite it as:
    • x+y=2a+2b+2=2(a+b+1)x+y = 2a+2b+2 = 2(a+b+1)
  • As aa and bb are integers, a+b+1a+b+1 is also an integer.
  • This means x+yx+y is an even number, for all possible odd values of xx and yy.
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Final statement

  • The sum of two odd numbers is an even number.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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