Set Theory - a MaS Tutor.

A set is a collection of objects, elements or members. The objects can be any collection you can think of. This collection can be described by listing the elements, or by giving a rule. The list or rule is given in a pair of curly brackets {}. For example: $A = {6, 9, 14}$ or $B = {x : x^{2} + 5 | x \in ℕ}$

The number of elements in the a set is given by $n (A)$ . For example: $n (A) = 3$ . The size of a set can be infinite.

∈ means is an element of;
∉ means is not an element of.
For example: $6 \in A$ or $5 \notin B$

The complement of a set $A$ is the set of all the elements not in $A$ and denoted by $A^{'}$

The empty set is denoted by Ø or {} and contains no elements.

The universal set is denoted by ℰ and contains all the elements of interest. This is how we show the universal set on a Venn diagram.

If a set $A$ is completely overlaps by another set $B$ then $A$ is a sub set of $B$ , $A \subset B$ . $B$ is a proper subset of $A$ if $n (B) < n (A)$ .

Set $A$ is equal to $B$ , $A = B$ if and only if both contain exactly the same elements. So, ${2, 3, 5} = {3, 5, 2}$

Where $A$ and $B$ overlap then $A$ intersects $B$ , $A \cap B$ , or $A and B$ . The elements listed by the intersection are on both sets.

$A \cap B = B \cap A$

Where we a looking for the members of two sets then this denoted by $A$ union $B$ , $A \cup B$ , or $A or B$ . Here or is inclusive, that is, the union lists elements in one or the other set, or both.

n (A \cup B) = n (A) + n (B) - n (A \cap B)

De Morgan's Laws

{(A \cup B)}^{'} = A^{'} \cap B^{'}

{(A \cap B)}^{'} = A^{'} \cup B^{'}

Set Notation Video (care of corbettmaths.com)
Venn Diagrams (care of www.mathsgenie.co.uk)
Set notation revision (care of mmerevise.co.uk)

GCSE Set Theory 101

Set (mathematics) according to en.wikipedia.org (Simple English Version)