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: $A=\{6,9,14\}$ 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, $\{2,3,5\}=\{3,5,2\}$

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)
• 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)

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Questions:

... serch for set notation questions ...
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Venn Diagrams

The relationship between different sets can be illustrated by using Venn diagrams (as above). Each set is represented by an enclosing loop e.g. circle or oval, contained within a further loop, usually a rectangle. This rectangle is called the universal set and contains all the elements of interest. If two sets contain some of the same elements the there will be an overlap. The overlap contains the shared elements. It could be that one set is contained completely within another set. There can sets which don't share elements in which case there will be no overlap.

Venn Diagrams

Practice Questions [MS];

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• Venn Diagrams (care of thirdspacelearning.com)
• Venn Diagrams (care of mmerevise.co.uk)
• Venn Diagrams (care of www.mathsgenie.co.uk)

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Common Numbers Sets and their Symbols

Here are the names and symobols of common number sets:

Set Name	Set Symbol	Description
Natural Numbers (or counting numbers)	ℕ	The positive whole numbers $\{1,2,3,4,...\}$ See also Natural Numbers (Care of simple.wikipedia.org)
Integers	ℤ	Whole numbers, positive or negative including zero. $\{...-3,-2,-1,0,2,3,...\}$
Rationals	ℚ	Numbers that are a ratio of two imegers $\{p/q:p\in \mathbb{Z},q\in \mathbb{N}\}$
Real Numbers	ℝ	Any value of the number line: They can be Positive, Negative or Zero; or Rational or Irrational; or be Algebraic or Transcendental.
Imaginary Numbers	𝕀	$ai$ where $a\in ℝ$ and $i^2=-1$
Complex Numbers	ℂ	Numbers of the form $a+bi$ where $a,b\in ℝ$ and $i^2=-1$

Natural numbers are a subset of Integers;
Integers are a subset of Rational Numbers;
Rational Numbers are a subset of the Real Numbers

See also:

Common Number Sets
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