Proof (DP IB Analysis & Approaches (AA): SL): Revision Note

Language of proof

 What is proof?

• Proof is a series of logical steps which show that a result (statement) is true for all specified numbers

  • It is not enough to prove a statement by testing just a few numbers

• Algebra is commonly used to help prove statements

What notation do I need to know?

• You need to be familiar with the following notation

  • $LHS$ is left-hand side

  • $RHS$ is right-hand side

  • $\equiv$ is the identity symbol

    • The LHS equals the RHS for all values of $x$

    • e.g. $x+x\equiv 2x$ (there's nothing to 'solve')

    • e.g. $-x\left(1-x\right)\equiv -x+x^2$

  • $\mathrm{\mathbb{Z}}$ is the set of integers $\left\{0,\pm 1,\pm 2,\pm 3,...\right\}$

    • ${\mathrm{\mathbb{Z}}}^{+}$ is the set of positive integers $\left\{1,2,3,...\right\}$

  • $\mathrm{\mathbb{N}}$ is the set of natural numbers

    • $\left\{0,1,2,3,...\right\}$

    • Zero is included, unlike ${\mathrm{\mathbb{Z}}}^{+}$

  • $\mathrm{\mathbb{Q}}$ is the set of rational numbers

    • These are numbers in the form $\frac{a}{b}$ where $a,b\in \mathrm{\mathbb{Z}}$ and $b\ne 0$

    • e.g. $\frac{2}{3},8\left(=\frac{8}{1}\right),-\frac{1}{7},0=\left(\frac{0}{1}\right),...$

    • ${\mathrm{\mathbb{Q}}}^{+}$ is the set of positive rational numbers

  • $\mathrm{ℝ}$ is the set of real numbers

    • ${\mathrm{ℝ}}^{+}$ is the set of positive real numbers, $\left\{x|x\in \mathrm{ℝ},x>0\right\}$

Proof by deduction

What is proof by deduction?

• Proof by deduction means proving that a statement using common mathematical results

  • For example

    • using algebra

    • using geometry

How do I apply proof by deduction to integers?

• To prove results about integers (whole numbers), you need to first represent the integers as algebraic letters or terms

  • The following table shows the most commonly used algebraic terms

• You then need to be able to apply operations to the terms above

  • Common operations are the

    • sum ($+$)

    • difference ($-$)

    • product ($\times$)

    • square ${\left(...\right)}^2$

How do I show that a result is odd or even?

• To prove an expression is even, show that it can be written as $2\times \left(\mathrm{integer}\right)$

  • For example, $2\left(n^2-3n\right)$ is even

    • This may require factorising out a 2

• To prove something is odd, show that it can be written as $2\times \left(\mathrm{integer}\right)+1$

  • For example, $2\left(n+m\right)+1$ is odd

• Make sure the part inside the brackets is an integer

  • For example, $2\left(n+\frac{1}{3}\right)$ is not even as $\frac{1}{3}$ is not an integer

• You can apply similar ideas to prove expressions are multiples of other numbers

  • For example, $7\left(n^2+2n\right)$ is a multiple of 7