3.02 Domain and range and set notation

Set notation

A set is a collection of objects that have a common property.  

So, if you think about all the things in your pencil case, this could be considered a set. 

You can probably come up with lots of other sets of objects. The clothes you have in your wardrobe, the names of streets you pass by on your way to school, animals that have four legs or people in your family for example. 

To describe a set using mathematical notation, we use large curly parentheses and list all the items of the set between them. Each object in the set is called an element. 

$\left\{\text{pencils, pens, sharpener, protractor, scissors, eraser, compass, glue, highlighter, calculator}\right\}${pencils, pens, sharpener, protractor, scissors, eraser, compass, glue, highlighter, calculator} 

The mathematical convention is to use capital letters when referring to the set, and lower case letters for elements in the set.  So the set $A$A of things I eat for breakfast can be written 

$A=\left\{\text{cereal},\text{eggs},\text{toast},\text{muffins}\right\}$A={cereal,eggs,toast,muffins}

and I would refer the element $a$a that is in $A.$A.

We also use the symbol $\in$∈ to make statements about whether elements are part of the set or not.

For example, $\text{toast}\in A$toast∈A and $\text{eggs}\in A$eggs∈A reads as toast is an element in the set $A$A and eggs is in the set $A$A. But if the element is NOT in the set then we use the symbol $\notin$∉ instead. So $\text{peaches}\notin A$peaches∉A and $\text{yogurt}\notin A$yogurt∉A reads as peaches are not in the set $A$A and yogurt is not an element in the set $A$A.

Of course we can have sets in mathematics as well, and these sets tend to have numbers or algebraic symbols.  

 

Finite sets

Let's have a look at some numerical sets. 

The set of odd numbers less than $10$10 would look like this: $\left\{1,3,5,7,9\right\}${1,3,5,7,9}

The set of multiples of $5$5 up to $50$50: $\left\{5,10,15,20,25,30,35,40,45,50\right\}${5,10,15,20,25,30,35,40,45,50}

The set of factors of $24$24: $\left\{1,2,3,4,6,8,12,24\right\}${1,2,3,4,6,8,12,24}

Some sets can just be groups of numbers that appear to have nothing else in common except that they are in the same set together. For example, $\left\{3,7.4,1004,33^4\right\}${3,7.4,1004,334}

These sets are called finite sets as they all have a finite number of elements.  The number of elements in a set is also called the sets cardinality, or the set's order.

 

Infinite sets

Here as some larger sets, 

The set of even positive integers: $\left\{2,4,6,8,...\right\}${2,4,6,8,...}

The set of multiples of $7$7: $\left\{7,14,21,28,...\right\}${7,14,21,28,...}

Or this set: $\left\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},...\right\}${12​,13​,14​,15​,...}

These sets are all infinite sets as the number of elements in them is infinite. The $3$3 dots at the end of each, called an ellipsis, indicates that the elements of the set continue on.

We can also use an ellipsis to save us from having to write all the elements in the middle of a set.  For example the set of positive integers up to $100$100 could be written like this, $\left\{1,2,3,...,99,100\right\}${1,2,3,...,99,100}.

 

Special Sets

Universal set: The set of everything relevant to the question is called the universal set. For your work in school mathematics involving sets the universal sets you will use most often is the set of integers or the set of real numbers.  

Empty set: Also called the null set, an empty set is (as you might guess) empty. It is a set that has no elements in it. It's important to note here that an empty set does not have a zero in it, it is completely empty!  We can write $\left\{\ \right\}${ } to represent the empty set, but there is also a special symbol we use to denote the empty set: $\varnothing$∅.

Equal sets: Two sets are equal if and only if they contain exactly the same elements. They can be equal even if the notation used to describe them is different. For example,

$A=\left\{2,3,6,11\right\}$A={2,3,6,11}and $B=\left\{11,2,6,3\right\}$B={11,2,6,3} then $A=B$A=B

$A=\text{set of first 5 primes}$A=set of first 5 primes and $B=\left\{2,3,5,7,11\right\}$B={2,3,5,7,11} then $A=B$A=B

 

Worked Examples

question 1

question 2

 

Domain and range

These two words sound similar, don't they? They both seem to be talking about areas and spans, and in math, they have similar definitions:

There are a number of ways to find the domain and range of a relation.  One way is to look at the coordinates given and simply list the possible values.

Consider the relation described by the set of points $\left\{\left(1,2\right),\left(5,3\right),\left(2,-7\right),\left(5,-1\right)\right\}${(1,2),(5,3),(2,−7),(5,−1)}. For this relation, the domain is $\left\{1,5,2\right\}${1,5,2} and the range is $\left\{2,3,-7,-1\right\}${2,3,−7,−1}. Notice how repeated values are not included and order is not important, as we only care about the possible values of $x$x and $y$y. 

The other method is to look at a relation graphically, and see how 'wide' or 'long' the graph is:

Horizontally this graph spans from $-1$−1 to $1$1, so we can write the domain as $-1\le x\le1$−1≤x≤1. Similarly, the graph goes vertically from $-2$−2 to $2$2 so the range can be written as $-2\le y\le2$−2≤y≤2.

 

Practice questions

Question 3

Question 4

Question 5