Lesson

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 heaps 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 brackets 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 $\text{yoghurt}\notin A$yoghurt∉`A` reads as peaches are not in the set $A$`A` and yoghurt 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.

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 sets order.

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}.

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 reals.

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: $\varnothing$∅.

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.

$A=\left\{2,3,6,11\right\}$`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`

We haven't yet finished with all the new terminology to work with sets.

- Imagine we have the set of all real numbers.
- Now imagine we take just the integers from that set.
- Now we take just the positive integers
- and then we take just the set of integers between $5$5 and $10$10.

What we have described here are subsets.

We define a subset as: $A$`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 $\subset$⊂. So $A\subset B$`A`⊂`B` is read as $A$`A` is a proper subset of $B$`B`. ($\not\subset$⊄ is used to say the opposite)

$A$`A` is the set of even integers and $B$`B` is the set of integers then $A\subset B$`A`⊂`B`.

$A=\left\{2,11,67,344\right\}$`A`={2,11,67,344}, and $B=\left\{1,2,8,11,67,120,180,344\right\}$`B`={1,2,8,11,67,120,180,344}, then $A\subset B$`A`⊂`B`.

$M=\left\{3,6,7,8\right\}$`M`={3,6,7,8} and $N=\left\{3,6,7,8\right\}$`N`={3,6,7,8}, then $M\subseteq N$`M`⊆`N`.

Not quite finished yet!

Let's consider the following. From the universal set of numbers $1$1 through $10$10, we define a subset $K=\left\{\text{even numbers}\right\}$`K`={even numbers}.

We could consider the image here as a representation of this situation.

Note how the universal set is depicted using the external rectangle, and our set inside uses a circle. This is the most common representation and lead us to Venn Diagrams which we will look at later.

What if we wanted to describe the elements from the universal set that are not included in the subset $K$`K`?

Well we could define a new subset and nominate the elements not in $K$`K` to be part of it, or we could use the notation $K'$`K`′ or $\overline{K}$`K` as this refers to all elements NOT IN $K$`K`.

The sets $U=\left\{20,8,26,3,15\right\}$`U`={20,8,26,3,15} and $V=\left\{20,8,26,3,15,2,24,10,27\right\}$`V`={20,8,26,3,15,2,24,10,27} are such that there are no other elements outside of these two sets.

Is $U$

`U`a subset or proper subset of $V$`V`?A subset.

AA proper subset.

BState the cardinality of $U$

`U`.List the elements of $U'$

`U`′.List the elements of the universal set. State the elements on the same line, separated by a comma.

Which set is $V'$

`V`′?The set $\left\{20,8,26\right\}${20,8,26}.

AThe empty set $\varnothing$∅.

BThe set $\left\{20,26,3,15\right\}${20,26,3,15}.

C

Consider the interval $\left(8,9\right]$(8,9] and answer the following questions.

Express the interval in set-builder notation:

{$x$

`x`$\mid$∣$\editable{}$}Graph the interval on the number line.