Lecture2: Set Theory

SET THEORY

SET, Defined:

·        is a collection of well-defined distinct objects.

e.g.: computer set = {mouse, printer, keyboard, CPU, monitor}

NAMING A SET

·        A set can be named using any capital letter in the English alphabet except from U.

·        Each object in a set is called “elements”.

            e.g.: C = { mouse, printer, keyboard, CPU, monitor}

                     V = {a, e, i, o, u}

                     A = {1, 2, 3, 4, 5}

WAYS IN DESCRIBING A SET

1.)  Tabular/Roster method – each elements are listed, separated by commas and enclosed in a curly braces.

e.g.: C = { mouse, printer, keyboard, CPU, monitor}

         V = {a, e, i, o, u}

         A = {1, 2, 3, 4, 5}

2.)  Rule method – it makes use of the description {x|x is an element of…}, which read as “x such that x is an element of…”

e.g.: C = {x|x is an element of computer hardware}

         V = {x|x is an element of vowels in the English alphabet}

         A = {x|x is an element of the first five positive integers}

           

KINDS OF SET

1.)  Equal set – are sets containing exactly the same elements.

e.g.:    A = {1, 2, 3, 4, 5}

            B = {2, 4, 1, 3, 5}

            Therefore, A = B

2.)  Equivalent set – are sets with the same number of elements.

e.g.:    A = {1, 2, 3, 4, 5}

            B = {a, e, i, o, u}

            Therefore, A ~ B

3.)  Finite set – contains countable number of elements.

e.g.:    A = {a, e, i, o, u}

            B = { x|x is an element of months in a year}

4.)  Infinite set – contains uncountable number of elements.

e.g.:    Z = {… , -3, -2, -1, 0, 1, 2, 3, …}

            B = {x|x is an element of grains of rice}

5.)  Joint set – are sets with elements in common.

e.g.:    A = {1, 2, 3, 4, 5}

            B = {2, 4, 6, 8, 10}

            A and B are joint set for it contains element in common, 2 and 4.

6.)  Disjoint set – are sets with no elements in common.

e.g.:    A = { x|x is an element of even numbers}

            B = { x|x is an element of odd numbers}

            A and B are disjoint set for it contains no element in common.

7.)  Universal set – is the totality of elements under consideration denoted as U.

8.)  Subset – is a part of a given set.

e.g.:    A = {1, 2, 3}

            {1} Ì A                      {2} Ì A                      {3} Ì A

            {1, 2} Ì A                  {1, 3} Ì A                  {2, 3} Ì A

            {1, 2, 3} Ì A

9.)  Null  Æ or Empty {} set – a set containing no elements.

VENN DIAGRAM

·        is a picture representation of sets, developed by John Venn in 1880, which makes use of rectangle as the universal set and circles as its subsets.

OPERATIONS ON SET

1.)  Union (È) – is the combination of elements in two or more sets without repetition.

e.g.:    A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10}

            AÈB = {1, 2, 3, 4, 5, 6, 8, 10}

*some reference uses (Ú) “or” symbol

to denote Union.

2.)  Intersection (Ç) – are the element common in the given sets.

e.g.:    A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10}

            AÇB = {2, 4}

*some reference uses (Ù) “and” symbol

to denote Intersection.

3.)  Complement (‘) – these are elements found in the universal set but not on the given set.

e.g.:    U = {0, 1, 2, 3, 4, 5}

            A = {2, 3, 4}

            A’ = {0, 1, 5}

  

4.)  Difference ( – ) – these are elements found in a given set but not on the other set.

e.g.:    A = {1, 2, 3, 4, 5}

B = {2, 4, 6, 8, 10}

          A – B = {1, 3, 5}

          B – A = {6, 8, 10}

GENERAL RULES OF SET

1.)  Union

A È U = U                 A È Æ = A                A È A = A

2.)  Intersection

A Ç U = A                 A Ç Æ = Æ               A Ç A = A

3.)  Complement

A’ È U = U                A’ È A = U                A’ Ç A = Æ

4.)  Difference

A – B = B’ Ç A          B – A = A’ Ç B

 PROPERTIES/ LAWS OF SET

1.)  Commutative Law

A È B = B È A                                              A Ç B = B Ç A

2.)  Associative Law

A È (B È C) = (A È B) È C                       A Ç (B Ç C) = (A Ç B) Ç C

3.)  Distributive Law

A È (B Ç C) = (A È B) Ç (A È C)                        A Ç (B È C) = (A Ç B) È (A Ç C)           

4.)  Identity Law

A È Æ = A                                                    A Ç U = A

5.)  Inverse Law

A’ È A = U                                                    A’ Ç A = Æ

APPLICATION OF SET THEORY

Sample Problem:

In an excursion at Enchanted Kingdom, 80 students brought Sandwich, Drinks and Chips as follows:

50 students brought Sandwich (S)

30 students brought Drinks (D)

30 students brought Chips (C)

18 students brought Chips and Drinks

15 students brought Sandwich and Chips

8 students brought Sandwich and Drinks

6

5 students brought all the said stuffs.

  

Construct a Venn Diagram and answer the following questions:

How many students brought:

1.)  Sandwich only? Drinks only? Chips only?       32;      9;       2

2.)  Nothing?                                                                   6

3.)  Sandwich and chips but not drinks?                          10      

4.)  Chips or drinks but not sandwich?                            24

5.)  Exactly one of the stuffs?                                         43

6.)  At most two of the stuffs?                                        75

7.)  At least two of the stuffs?                                         31