# Set Theory

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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: $A=\left\{6,9,14\right\}$ or $B=\left\{x:{x}^{2}+5|x\in ℕ\right\}$

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.

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

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

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.

$A∩B=B∩A$

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.

$n ( A∪B ) = n ( A ) + n ( B ) - n ( A∩B )$

## De Morgan's Laws

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

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

GCSE Set Theory 101