## Notation:

In mathematics, a set is a collection of objects (in our case numbers). We can write a set using curly brackets, for example $$\{1, 5, 7\}$$.

We say something is an 'element' of a set if it is contained in it, for example 1 is an element of the set $$\{1, 5, 7\}$$. We write $$1\in\{1, 5, 7\}$$ to mean this. Similarly, we can use the $$\notin$$ symbol to mean that something is not an element of a set. The empty set is one with no elements, and it is denoted by the symbol $$\emptyset$$.

If a set $$B$$ is a subset of $$A$$, then all of the elements of $$B$$ are also elements of $$A$$. We write $$B\subset A$$ to mean this. The union of two sets is the set that contains all the elements of both sets, written as $$A \cup B$$. The intersection of two sets is the set containing only the elements that are contained in both sets, written as $$A \cap B$$.

## Classification of numbers:

For A-Level, you should know basic sets of numbers in set notation:

 bbb "N" Natural numbers - the set of all positive integers: $$\{1,2,3,4,5,\ldots \}$$ (different definitions may include $$0$$, there is no definite answer as to whether it should or shouldn't be included). bbb "Z" Integers - the set $$\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$$ bbb "Q" Rational numbers - those which can be expressed as the quotient of two integers, i.e. in the form $$\frac{a}{b}$$ where a, b in bbb"Z" . bbb "I" Irrational numbers - numbers that are not rational. bbb "R" Real numbers - numbers that can represent a distance along a line.

Where bbb "N" sub bbb "Z" sub bbb "Q" sub bbb "R"

## Expressing ranges in set-builder notation:

You may be asked to express a range in set notation. For example, if we had the range $$0<x<1$$ for x in bbb "R", we would write this as:

{x in bbb "R" | 0<x<1}

Where the x in bbb "R" indicated that we are working in real numbers, and the vertical bar "$$|$$" (sometimes "$$:$$" is used instead) means "such that". I.e. the set notation says "x is an element of reals such that $$0<x<1$$".

If you wanted to write a more complicated range, such as $$0<x<1$$ or $$x>2$$, you could write:

{x in bbb "R"| 0<x<1}uu{x in bbb "R"| x>2}