Proof & Reasoning (DP IB Analysis & Approaches (AA))

Proof in mathematics is the use of a series of logical steps to show that a mathematical result is true for all specified cases.

E.g. showing that the square of an odd number is odd for every odd number.

False.

Seeing that a result works for a few numbers is not enough to show it will work for all numbers.

A valid proof is required to show that it works for all values.

The set of all integers is denoted by the symbol .

The set of all positive integers (1, 2, 3, ...) is denoted by the symbol .

The symbol denotes the set of all natural numbers (0, 1, 2, 3, ...).

is the symbol for the set of all rational numbers.

I.e., the set of all numbers that can be written in the form , where and are integers and .

A mathematical identity is a statement that is true for all possible values of any variables used (for example or ).

A mathematical identity is denoted by the symbol .

A proof by deduction uses known facts, and logical steps and arguments connecting those facts, to show that a mathematical result is true.

In proofs, even numbers are typically represented as , where is an integer.

( or are also often used in this way.)