1.1.2 Proof by Deduction

Proof

Proof is a mathematical tool used to test whether statements are true or not. There are many different types/styles of proof that can be used in different situations.

Proof by Deduction

This is a type of proof used to deduce other true statements from statements we already know to be true.

Notation (⇒)

This sign means "implies that". It means that the statement on the right of the sign is a direct consequence of the statement on the left.

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(i)

• $P: x^2+x−2=0$

• $Q: x=1$

• Solution:

  • $(x + 2)(x - 1) = 0 \Rightarrow x = -2 \text{ or } 1$
  • $\therefore P \Leftarrow Q$ (ii)

• $P: y^3 > 1$

• $Q: y > 1$

• Solution:

  • $∴P⇒Q$

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(i) $n^3 + 1 \text{ is an odd integer} \quad \boxed{\Rightarrow} \quad n \text{ is an even integer}$

$⇒ n$ being even means $n^3$ (EVEN×EVEN×EVEN) is also even. Add 1 then it becomes odd.

$\not\Rightarrow$ because letting $n^3+1=7$, we get $n^3=6$ and $n = \sqrt[3]{6}$ which is not even.

(ii) $(x - 3)(x - 2) > 0 \quad \boxed{\Leftarrow} \quad x > 3$

$\Leftarrow$ because from the graph we can see that $x>3$ or $x<2$ satisfy the inequality. Not just $x>3$.

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Proof Involving Odd and Even Numbers

An even number is any integer multiple of 2 (including 0). An odd number is one that is not even (but must be an integer).

Mathematical Definitions

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Prove that the sum of the squares of any two consecutive numbers is odd.

Let my two numbers be $n,n+1$

Used the same letter as it is important in the question that numbers are consecutive

$n^2 + (n+1)^2$

$=n^2+(n=1)(n-1)$

$= n^2 + (n^2 + 2n + 1)$

$= 2n^2 + 2n + 1$

$= 2(n^2 + n) + 1$

Take out a factor of 2 from wherever we can.

Which is of the form $2A+1, \therefore$ odd.

Prove that the product of two odd numbers is also odd.

1. Define your starting numbers:

• Let $(2m+1)$ and (2n+1) be the two odd numbers in question.

2. Perform whatever operation was requested:

• $(2m+1)(2n+1) = 4mn + 2n + 2m + 1$

3. Factorize in a way that proves evenness or oddness:

• $2(2mn + n + m) + 1$
• Emphasize by leaving $+1$ outside the bracket.

4. Conclude by spelling out what you have just shown:

• Since this is of the form $2A + 1$, we can deduce the product of two odd numbers is odd.

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Notation

• ⇒ means "implies that" or "leads to the direct consequence"

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5. (i)

• $P:$ $x^2 + x - 2 = 0$
• $Q$ : $x = 1$
• Solution:
• $(x + 2)(x - 1) = 0 \Rightarrow x = -2, x = 1$
• $\therefore P \Leftarrow Q$

6. (ii)

• $P :$ $y^3 > 1$
• $Q$ : $y > 1$
• Solution:
• $P \Leftrightarrow Q$

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