IB Maths AA 1.6 Notes

Proofs & key notation

A proof is a sequence of logically valid steps showing that a statement is true. Here, the focus is on simple deductive proof, including numerical and algebraic proofs, and on setting out a proof clearly from the left-hand side to the right-hand side.

It is important to distinguish between an equation and an identity.

• An equation is true only for particular values of the variable. For example, $2x+1=7$ is true only when $x=3$.
• An identity is true for all values of the variable for which both sides are defined. For example, $(x-3)^2+5\equiv x^2-6x+14$ is an identity because both expressions are always equal.

The symbol $=$ means equal. The symbol $\equiv$ means identically equal.

You should be familiar with the following notation:

• $LHS$ means left-hand side
• $RHS$ means right-hand side
• $\equiv$ means identity
• $\implies$ means implies
• $\iff$ means if and only if, or equivalently
• $\therefore$ means therefore

Useful number sets include:

• $\mathbb{N}$ for natural numbers
• $\mathbb{Z}$ for integers
• $\mathbb{Q}$ for rational numbers
• $\mathbb{R}$ for real numbers