19.01 Language and notation of sets

Before we calculate probabilities, let's familiarise ourselves with the language and notation for describing components in this topic. In probability, the sample space is a list of all the possible outcomes of an experiment.

Outcomes are the results of an experiment or trial. For example, think about flipping a coin. There are two possible outcomes - a head or a tail. So when we list (or write out) the sample space, we write:  

$S=\left\{heads,tails\right\}$S={heads,tails}

We can write a sample space using a list, table, set notation as above or a diagram such as a Venn or Tree diagram.

An event is a subset of the of the sample space and is often represented by a capital letter to abbreviate the description of the event in calculations. For example we could have the experiment of rolling a six-sided dice and let $E=\text{the event of rolling an even number}$E=the event of rolling an even number, $O=\text{the event of rolling an odd number}$O=the event of rolling an odd number and $A=\text{the event of rolling a number less than 3}$A=the event of rolling a number less than 3.

Then our sample space is $S=\left\{1,2,3,4,5,6\right\}$S={1,2,3,4,5,6} and the events can be written as the sets $E=\left\{2,4,6\right\}$E={2,4,6}, $O=\left\{1,3,5\right\}$O={1,3,5} and $A=\left\{1,2\right\}$A={1,2}.

Notation for probability

The notations commonly used in probability are related to function notation and also to the notations of set theory. This is because the range of possible outcomes of an experiment or observation is a set. We assign numbers - called probabilities - to subsets of these outcomes in the manner of a function.

Thus, the notation $P\left(A\right)$P(A) means 'The probability that event $A$A occurs'. 

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 and for probability questions this is the sample space.
• 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 $A\cup B=4,5,11,12,15,16,17,20,25,30$A∪B=4,5,11,12,15,16,17,20,25,30

 

Complementary events

A complement of an event are all outcomes that are NOT the event. If $A$A is the event 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 $\left\{Heads\right\}${Heads}, the complement $A'$A′ is $\left\{\text{not a head}\right\}${not a head} which is $\left\{Tails\right\}${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 events

If events are mutually exclusive, it means they cannot happen at the same time. 

Some examples of experiments that involve mutually exclusive events are:

• Tossing a coin - Consider the events 'flipping a head' and 'flipping a tail'. You cannot flip a head and a tail at the same time. 
• Rolling a die - Consider the events 'Rolling an even number' and 'rolling an odd number'. We can't roll any number which is both even and odd. 
• Picking a card from a deck of cards - Consider the events 'Drawing a $7$7 card' and 'Drawing a $10$10 card'. They have no outcomes in common. There is no card that is both a $7$7 and a $10$10.

Since these events cannot both occur at the same time, they are mutually exclusive events.

However, some events can happen at the same time and we call this non-mutually exclusive. For example:

• Picking a card from a deck of cards - Consider the events 'drawing a Club card' and 'drawing a $7$7'. They have outcomes in common. We could pick a card that is a Club and a $7$7, because I could get the $7$7 of clubs. 
• Rolling a die - Consider the events 'Rolling an odd number' and 'Rolling a prime number'. They have outcomes in common, namely the numbers $3$3 and $5$5. 

Since these events can both occur at the same time,  they are not mutually exclusive events.

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, 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 questions

Question 1

Question 2

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question 5