1.09 Language and notation of sets

 

Set notation

As we are commonly going to use set notation, let's review some terms and properties of sets:

• Each object in the set is called an element. 
• We also use the symbol $\in$∈ to make statements about whether elements are part of the set or not. For example, $2\in\left\{1,2,3\right\}$2∈{1,2,3} and we use the symbol $\notin$∉ to indicate if something is not an element of the set. So $4\notin\left\{1,2,3\right\}$4∉{1,2,3}
• Finite sets have a finite number of elements.  The number of elements in a set is also called the cardinality of the set, or order. For example the set of odd numbers between $2$2 and $8$8 is the finite set $\left\{3,5,7\right\}${3,5,7} and has cardinality $3$3.
• Infinite sets have an infinite number of elements. Examples could be the set of positive even numbers $\left\{2,4,6,8,...\right\}${2,4,6,8,...} the set of numbers on the interval between $1$1 and $5$5 that is $\left\{x\ |\ 1\le x\le5\right\}${x | 1≤x≤5}. What is the cardinality of these sets? Are they the same size?
• The set of everything relevant to the question is called the universal set.
• The empty set or the null set, is a set that has no elements in it. We can write $\left\{\ \right\}${ }to represent the empty set, but there is also a special symbol we use to denote the empty set: $\varnothing$∅.
•  $A$A is a subset of $B$B if and only if every element of $A$A is in $B$B.  We use the symbol $\subseteq$⊆ to describe subsets. So $A\subseteq B$A⊆B is read as $A$A is a subset of $B$B. We also have  the symbol $\not\subseteq$⊈, for the "not a subset of" statement. ) If there is at least one element in $B$B that is not included in the subset $A$A, then we call this a proper subset, and use the symbol $\subset$⊂.

 

Intersections

Just like how a road intersection is the place where two roads cross paths, an intersection of sets is where two sets overlap. Elements that appear in the intersection of sets are elements that have the same characteristic as both the individual sets. 

Mathematically we write the intersection of sets using the intersection symbol, $\cap$∩.  We interpret the intersection of $A$A and $B$B, $A\cap B$A∩B to be what appears in both set $A$A and set $B$B. It helps some students to relate $\cap$∩ to AND or to think of the symbol like a bridge joining both sets. 

For example

$A=\left\{5,10,15,20,25,30\right\}$A={5,10,15,20,25,30} and $B=\left\{6,12,18,24,30\right\}$B={6,12,18,24,30} then $A\cap B=\left\{30\right\}$A∩B={30}

Unions

If we consider the intersection the 'and' of mathematical sets, then the union is the OR.  $A\cup B$A∪B is the notation we use, and we would read this as the union of $A$A and $B$B. It is the set of the elements that are in either $A$A or $B$B.

For example

$A=\left\{5,11,16,17,20,25\right\}$A={5,11,16,17,20,25} and $B=\left\{4,12,15,25,30\right\}$B={4,12,15,25,30} then $A\cup B=\left\{4,5,11,12,15,16,17,20,25,30\right\}$A∪B={4,5,11,12,15,16,17,20,25,30}

 

Complementary sets

A complement of a set contains all the elements that are NOT in the set. If $A$A is the set then the complement is denoted $A'$A′ or sometimes $\overline{A}$A.

The following are examples of events and their complements:

• If event $A$A is tossing a coin and getting $\left\{\text{Heads}\right\}${Heads}, the complement $A'$A′ is $\left\{\text{Not a head}\right\}${Not a head} which is $\left\{\text{Tails}\right\}${Tails}
• If event $B$B is rolling a $6$6 sided die and getting $\left\{2\right\}${2}, then the complement $B'$B′ is $\left\{1,3,4,5,6\right\}${1,3,4,5,6}

Mutually exclusive sets

If sets are mutually exclusive, it means they do not have any elements in common. 

 

Venn diagrams

A Venn Diagram is a pictorial way to display relationships between different sets.  The idea of a Venn diagram was first introduced by John Venn in the late 1800's and they are still one of the most powerful visualisations for relationships.

Worked example 

Example 1

For the numbers between $2$2 and $20$20, let $E=\left\{\text{even numbers}\right\}$E={even numbers}, and $M=\left\{\text{multiplies of 3}\right\}$M={multiplies of 3}.  

Place the numbers in the appropriate sections, and for each ask "Is the number even? Is it a multiple of $3$3? Is it both or Is it none of those options?"

Take note of how the numbers that do not fit into either set are placed outside the circles, but still within the bounds of the universal set.

Using Venn Diagrams when solving problems about sets will need us to be able to identify using set notation the regions in the Venn Diagram. The following applet will let you explore the different regions.  

Created with Geogebra Originally created by Mathguru

 

Practice question

Question 1