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Set Theory

A Prezi for the better understanding of Set Theory for students, particularly Subsets, Equal Sets, Equivalent Sets, Finite & Infinite Sets and Venn Diagrams. Feel free to edit and reuse.
by

Daniyal Admaney

on 6 November 2014

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Transcript of Set Theory

SETS
Introduction to Sets
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s
A set is a well defined collection of distinct objects.
The objects that make up a set which are also known as the elements or members of a set can be anything from numbers to people to letters of the alphabet to other sets, and so on.
The items you wear daily like shoes, socks, hat, shirt, pants and so on can be classified as a set.
Sets are divided into many categories. 3 of them being: Venn Diagrams, Finite & Infinite Sets and Subsets & Equal Sets
Venn Diagrams
Subsets & Equal Sets
Finite & Infinite Sets
- A subset is a portion of a set.
- E.g: B is a subset of A, if every member of B is a member of A.
- A subset is further divided into 2 categories:
Proper Subsets
and
Improper Subsets.
SUBSETS
A
B
Proper Subsets
A proper subset A' of a set A, is a subset that is strictly contained in A and so necessarily excludes at least one member of A.
For example, consider a set {1,2,3,4,5}. Then {1,2,4} and {1} are proper subsets, while {1,2,6} and {1.2.3.4.5.6} are not proper sets.

A proper subset is denoted by this symbol.
Improper Subsets
An improper subset is defined as the subset consisting of all elements of a given set.
An improper subset is identical to the set of which it is a subset.
For example:
-Set A= {1, 2, 3, 4, 5}
-Set B= {1, 2, 3, 4, 5}
Therefore Set B is an improper subset of Set A and vice versa.
An improper set is denoted by:
EQUAL SETS
Equal sets are those who consist of exactly the same elements. Equal sets are always equivalent though not all equivalent sets are equal.
Two sets are equal if they both have the same members.
Example
If, F = {20, 60, 80}
And, G = {80, 60, 20}
Then, F=G, that is both sets are equal.
Note: The order in which the members of a set are written does not matter.
F=G
Venn diagram for Equal Sets
Equal Sets in real life?
- There are 3 friends: Ali, Daniyal and Farhad
- Assume Set A= people who use an iPhone,
while, Set B= people who like to play cricket.
- A={daniyal,farhad,ali} | B={farhad,ali,daniyal}
- Since both Set A and B contain the same
elements, they are equal sets.
- Therefore, Set A = Set B
EQUIVALENT SETS
In the set theory, equivalent sets are sets are those with the same amount of elements. Hence, they can have one to one correspondence with each other. Two sets are equivalent if they have the same number of elements.
Example
If, F = {2, 4, 6, 8, 10}
And, G = {10, 12, 18, 20, 22}
Then, n(F)= n(G)= 5, that is, sets F and G are equivalent.
Venn Diagram of Equivalent Sets
Use of equivalent sets in real life

Q: Difference between improper subsets and equal sets?

A: There is no difference between improper subsets and equal sets. If A is an improper subset of B then A = B which is the same as an equal set. This is the reason why the term 'improper subset' is rarely used.
Real life examples of Proper
and Improper Subsets
Finite Sets
Infinite Sets
A finite set has limited (finite) number of elements, which can be counted. For example, {2,4,6,8,10} is a finite set with five elements.
D = {days of the week}
= {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
And, n(D) = 7

Example of Finite Sets
H = {students who take History}
= {Alan, Brendon, Taylor, Rodney}
And, n(H) = 4
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are:
• The set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
• The set of all real numbers is an uncountably infinite set.

Examples of Infinite Sets
Maths=Art
Like we use art to express our words, we can also express numbers through art.
What?
A diagram using circles to represent sets, with the position and overlap of the circles indicating the relationships between the sets.
Who?
Venn Diagrams were introduced by John Venn(August 1834-April 1923)

His Venn Diagrams are used in many fields, including set theory, probability, logic, statistics, and computer science.
Venn diagrams normally comprise overlapping circles.
The interior of the circle symbolically represents the elements of the set, while the exterior represents elements that are not members of the set.
For instance, in a two-set Venn diagram, one circle may represent the group of all wooden objects, while another circle may represent the set of all tables.
The overlapping area or intersection would then represent the set of all wooden tables.
Wooden Dining Table
Glass Coffee Table
Wooden Objects
Table
Wooden Coffee Table
Wooden Side Table
Glass Computer Table
Cupboard
Chair
Cricket Bat
Book Shelf
Why?
Venn diagrams are used to represent sets because,
Graphic organizers are powerful ways to help students understand complex ideas;
Drawings and diagrams engage visual learners;
They show relationships, clarify concepts, and facilitate communication.
There are 3 types of functions of a Venn Diagram:
1. Union
2. Intersection
3. Difference
The following diagram(s) will clear the difference
between them:
Intersection
U
Union
Difference
You could have a set made up of your ten best friends:
{alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}

Each friend is an "element" (or "member") of the set (it is normal to use lowercase letters for them

Now let's say that Alex, Casey, Drew and Hunter play Soccer:
Soccer = {alex, casey, drew, hunter}

And Casey, Drew and Jade play Tennis:
Tennis = {casey, drew, jade}
We could put their names in two separate circles:


Soccer
alex, casey,
drew,
hunter
Tennis
casey,
drew,
jade
U
U

You can now list your friends that play Soccer OR Tennis.
This is called a "Union" of sets and has the special symbol:∪

Soccer Tennis = {alex, casey, drew, hunter, jade}

Not everyone is in that set, only your friends that play Soccer or Tennis (or both).
We can also put it in a "Venn Diagram":



A Venn Diagram is clever because it shows lots of information:

It shows us that Alex,Hunter,Casey and Drew play Soccer
Also, Casey,Drew and Jade play Tennis
It shows us that Casey and Drew play both sports.
"Intersection" is when something is in BOTH sets.

In our case that means they play both Soccer AND Tennis; which is Casey and Drew.

The special symbol for Intersection is an upside down "U" like this: ∩

And this is how we write it down:
Soccer ∩ Tennis = {casey, drew}
U
Which Way Does That "U" Go?
You can also "subtract" one set from another.

For example, taking Soccer and subtracting Tennis means, people that play SOCCER but NOT TENNIS, which would be Alex and Hunter.

The symbol for it is the minus sign:

And this is how we write it down:
Soccer Tennis = {alex, hunter}
Venn Diagram: Difference of 2 Sets
∪ is Union: is in either set
∩ is Intersection: must be in both sets
is Difference: in one set but not the other
U
U
-
-
-
Summary
3 Set Venn Diagrams
You can also use Venn Diagrams for 3 sets.

Let us say the third set is "Volleyball", which drew, glen and jade play:
Volleyball = {drew, glen, jade}
The Venn Diagram is now like this:
U
U
You can see (for example) that:

Drew plays Soccer, Tennis and Volleyball
Jade plays Tennis and Volleyball
Alex and Hunter play Soccer, but don't play Tennis or Volleyball
no-one plays only Tennis
Set S

Union of Sets T and V

U
Intersection of Sets S and V

U
Union of 3 Sets: S ∪ T ∪ V
S ∩ V = {drew}
S = {alex, casey, drew, hunter}
T ∪ V = {casey, drew, jade, glen}
Intersection of Sets S and V, minus Set T
(S ∩ V) T = { }
U
-
The { } implies that it is an Empty Set, a set without elements
Venn Diagrams also consist of two more types of Sets:
Universal Set
and
Compliment of a Set
The Universal Set is the set that contains everything. Well, not exactly everything; everything that we are interested in now.
The symbol used for the Universal Set is the lowercase
of the Greek Alphabet, Xi:
In our case the Universal Set are our Ten Best Friends.
= {alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}
We can show the Universal Set in a Venn Diagram by putting a box around the whole thing:

Now you can see ALL your ten best friends, neatly sorted into what sport they play.
We can do interesting things like take the universal set and subtract the ones who play Soccer:
We write it this way:
S = {blair, erin, francis, glen, ira, jade}
Which says: The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}
In other words: everyone who does not play Soccer.
Universal Set
-
Compliment of a Set
The Compliment of a Set is everything except that particular set.

It is a special way of saying "everything that is not"

We show it by writing an apostrophe ( ) like this:

We refer to it as: "S(the name of the set) Compliment"
'
'
S
= {blair, erin, francis, glen, ira, jade}
S
'
-
(just like the example, before!)
S
Summary
is the Complement of Set A: everything that is not in A
Empty Set: the set with no elements. Shown by { }
Universal Set ( ): all things we are interested in
A
'
Common example of Infinite Sets consist of:
The stars in our galaxy
The hair on our head
2
visualizations
QUESTION/ANSWERS SESSION
APPLICATION
GUIDED PRACTICE
THANK YOU
Daniyal Admaney - Farhad Ahmad - Ali Shamim
Full transcript