Beginning Mathematics/Set Theory

In this section we shall go over some basic set theory.

Sets

A set is a collection of elements, we note that the elements may not be numbers. Elements of sets may be letters, objects, or even other sets. Sets can be finite, or infinite. Often sets will have a pattern of elements or the elements will be similar in some fashion. Sets are denoted by curly braces ${\displaystyle }$. The elements of the set are indicated between the braces. At times, when it should be clear what the pattern is, the set will end with "..." indicating the elements continue in that fashion. Other times the set may be described in words or mathematical notation. Mathematical notation is read as in the following example:

$\{x$ | x has a nice property $\}$

This is read as "The set of x such that x has a nice property". Here the | stands for "such that" often a colon (:) will be used instead.

To denote an element is in a set we write: $x\in A$ this reads "x is an element of the set A" or "x is in A".

Special Sets

Some sets are used frequently in mathematics so they are given by special notation.

$\mathbb{N}=𝒻Setofwholenumbersℊ=\{0,1,2,3,4,5,6,...\}$

$\mathbb{Z}=\{Setofallintegers\}=\{...,-3,-2,-1,0,1,2,3,...\}$

$\mathbb{Q}=\{Setofrationalnumbers\}=\{\frac{a}{b}𝒿a,b\in \mathbb{Z}andb\ne 0\}$

$ℝ=\{Setofrealnumbers\}$

$ℂ=\{Setofcomplexnumbers\}=\{a+bi𝒿a,b\in ℝandherei=\sqrt{-1}\}$

Examples

$𝒻1,2,3\}$

$𝒻7,12,80\}$

$\{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}=lettersofenglishabc$

$𝒻1,2,3,4,5,...\}=\{Setofpositiveintegers\}$

$\{2,4,6,8,10,12,14...\}=\{$ Set of positive even integers $\}=\{x|x\in \mathbb{Z}$ and x is even $\}$

Basic operations

Union

The union of two sets is denoted by $\bigcup$. The union of two sets is best described as "everything in both the sets once".

Formally: $A\bigcup B=\{x|x\in Aorx\in B\}$

For example:

$A=𝒻1,2,3\}$ $B=𝒻1,2,3,4,5,6\}$

$A\bigcup B=\{1,2,3\}\bigcup \{4,5,6\}=\{1,2,3,4,5,6\}$

Intersection

The intersection of two sets is denoted by $\bigcap$. The intersection of two sets is best described as "only what is in both of the sets".

Formally: $A\bigcap B=\{x|x\in Aandx\in B\}$

For example:

$A=𝒻1,2,3,4,5,6\}$ $B=𝒻2,4,6,8,10\}$

$A\bigcap B=\{1,2,3,4,5,6\}\bigcap \{2,4,6,8,10\}=\{2,4,6\}$