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Set Theory

In this unit we will explore a the branch of mathematics known as Set Theory. We will focus on basic symbols, notations, terminology,set operations and Venn Diagrams. However, set theory contains much more information and can even be the focus of graduate level math studies. Most mathematicians regard Set theory as the foundation of mathematics because everything can be tied into the concept of Set Theory. It is used throughout mathematics from from fundamental concepts to statistics, probability,advanced modern mathematics,etc.
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What is a Set?

Real Word Examples
If you are a tennis fan then you know one of the biggest tennis events in the world is the US Open. If you watch a match you will often hear something like "Federer is up a set".
tennis In tennis, what is the meaning of a set?


Answer
What other real world uses of 'sets' can you think of ?


Answer
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What is a set in mathematics?

the technical definition
In mathematics, we define a set as a of well defined, unordered, distinct objects or elements
These can be anything: numbers, people, animals, symbols, letters or even other sets. Sets are conventionally denoted with capital letters and elements are enclosed by curley brackets.

Examples

$ A = \{2,4,5,8, 10\} \\ B = \{Horse, Cat, Dog, Rabbit \} \\ C = \{ Giants, Jets, Cowboys, Eagles \} \\ S = \{ \{1,2 \} , \{3,4,5\} , \{ A\}, \{ B\} \} $
(Set S is a set whose elements themselves are sets. )
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When are two sets equal?

It's all about being'distinct'
If you look carefully at the definitio of a set, you will notice the word distinct This is a fancy way of saying that if elements ar repeated, they still only count once.
For example, consider the following two sets, Set A and Set B:
$ A = \{ 2, 4, 6, 8 \} \\ B = \{ 2, 2, 2, 4,4, 6, 6, 8 \} $
Since we only care about distinct elements, set B is really just $$ \{2,4,6,8\}$$.

Therefore, A = B

Order

The definition of set states that a "set is an unordered collection of elements. What does this mean about sets C and D, below?
$ C = \{ 10, 27, 4 \} \\ D = \{ 4,10,27 \} $
Answer
Sets C and D are equal because there elements are the same. The only difference is the order, which doesn't matter.

Practice Identify Equal Sets

Directions : Look at each pair of sets below and determine if they are equal
1) A= {-1,0,1,2 } and B= {-1,0,1 } Equal
Not Equal
2) C = {10, 20, 30, 40} and D = {40, 30,20,10} Equal
Not Equal
3) E= { red, red, blue, green, blue} and F= {red,green,blue} Equal
Not Equal
4) G = {a, e, i, o, u} and H = { e,e,a,o,u } Equal
Not Equal
5) I = { Queen, Pawn, Knight , Bishop} and J = { Bishop, Pawn, Queen, Rook} Equal
Not Equal
6) I = { ♥ , ♠, ♦, , ♣ } and J = { ♥ , ♠, ♦, , ♣ } Equal
Not Equal

Elements of a set

Given the following two sets:
$ J = \{2, 4, 5, 10, 14 \} \\ K = \{ \pi , \sqrt{3} , -2\pi, - \sqrt{5} , \frac{2\pi}{3} \} $
If we want to say that "2 is an element in Set J", but that "$\pi$ is not an element of Set J", we have a simplie mathmeathical notation to concisely write this:
Set Notation Meaning
$2{\in} J $ "2 is an element in set J"
$2 {\notin} J $ "2 is not an element in set J"

Practice With Set Notation

Directions :The questions below refer to sets J and K above. Choose the correct symbol that goes in the blank.
1) 5 J
2) 2$\pi$ k
3) 12 J
4) $\sqrt{3}$ K
5) $\sqrt{5}$
6) 6 J