A set is just a collection of objects, that have some quality in common, or that can be described by some rule. Although in maths we tend to think of sets as groups of numbers, they do not have to be.

For example

{Pure, Mechanics, Statistics, Discrete}

 would be the set of components of A-level maths.

We show that they are a set by writing them in curly brackets.

It is not necessary to list all the members of a set. You could instead just give the rule that defines the set. Suppose that we name our set M;then

M={Components of A-level Maths}

Clearly it is more convenient to describe a set this way if the set has lots of members

{n:n = integers from 0 to 1000}             would be extremely tedious to write out.

In some cases,  of course, it would be impossible to write out all the members, as they are infinite

{n : n = all integers}

There are some specific sets that you need to be aware of. Most of these also feature in functions, where they are used to describe Domains.

N = {Natural Numbers; the positive integers plus 0; 0,1,2,3,4,....}

Z = {integers;...,-2, -1, 0, 1, 2, ....}

Q  = { Rational Numbers; numbers that can be written m/n; m,n are integers and n is not 0}

R =  {Real Numbers; these include all rational and irrational numbers}

Unless otherwise stated, you can assume that you are dealing with Real Numbers.

You should also be aware of set terminology and notation.

To denote that a number is a member of a set, we say

We can also say that a number is not a member of a set

The Empty Set ( or Null Set)

This is the set that has no members. An example could be Natural Numbers less than 0. This set is denoted Ø.

Subsets

This is a set which is entirely contained within another set. For example the Natural Numbers are a subset of the Integers, because every Natural Number is also an Integer. In turn, the Integers are a subset of the Real Numbers.

Disjoint Sets

 These are sets that have no members in common. For example the even numbers and the odd numbers form disjoint sets.

The Universal Set E

 This is the set that contains everything that you are interested. All of the other sets that you deal with will therefore be subsets of the Universal Set. For Example a biologist might have a Universal Set of all living things. They would also have sets of Animals, Plants, Fungii, but these would all be subsets of Living Things.

Other Notation will be dealt with as it arises, when looking at Venn Diagrams.

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