Set Theory

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 $B=\{x:x^2+5|x\in \mathbb{N}\}$

The number of elements in the a set is given by $n\left(A\right)$. For example: $n\left(A\right)=3$. 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 $A^{\prime }$

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

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

If a set $A$ is completely overlapped by another set $B$ then $A$ is a sub set of $B$, $A\subset B$.

Where $A$ and $B$ overlap then $A$ intersects $B$, $A\cap B$, or $AandB$. 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 $AorB$. Here or is inclusive, that is, the union lists elements in one or the other set, or both.

$$n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B)$$

De Morgan's Laws

${(A\cup B)}^{\prime }=A^{\prime }\cap B^{\prime }$

${(A\cap B)}^{\prime }=A^{\prime }\cup B^{\prime }$

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

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GCSE Set Theory 101

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