W1_Sets

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Centre for Computer Technology

ICT114Mathematics for

Computing

Week 1

Introduction to Set Theory

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March 20, 2012 Copyright Box Hill Institute

Objectives

Set, Set Notation

Subsets

Basic Set Operations Venn Diagrams

Some important sets

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A simple problem

In a survey of 100 college students, the following wasreported regarding the number of students takingdifferent course: Physics 50, Computers 40.Mathematics 30, Physics and Computers 22, Physics

and Mathematics 18, Mathematics and Computers15, Physics, Mathematics and Computers 10.

(1). How many students did not take any of Physics,

Mathematics, Computers?

(2). How many students opted for Computers but notMathematics ?

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Set : Introduction

A set is a well-defined list, collection or class of objects.

The objects could be anything : numbers, names,

people, cities. These objects are called the elements ormembers of the set.

Example 1: The numbers 1,3,5,7,9,11,13,

Example 2: The solutions of the equation x2 4x+3=0

Example 3: The list of IP addresses in a subnet

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Notation

Sets are usually denoted by capital letters

A, B, P, X, ..

The elements are usually represented bylowercase letters a, b, p, x, ..

There are two forms for presentation of a set :Tabular form , A = {1,3,5,7,9,11,}

Set builder form, A = {x | x is odd}

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If an object x is a member of set A,

we write xA, which is usually read as

x belongs to A; or x is in A

If an object x is not a member of set A,

we write x A.

Example : Let B={ y | y is a perfect square}

4 B, 9 B,

But, 12 B, 23 B

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Some more concepts

Set A and set B are said to be equalif both havethe same members, irrespective of the order.We write A = B

Example 1 : X={1,2,3,4} is equal to Y={4,2,3,1},so X = Y

Example 2 : R = {1,2,3,4,5} is equal to

S = {x | x is a positive integer less than 6}

A set which does not contain any element iscalled the null setand is denoted by . It isalso called voidor empty

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Subsets

If every element in a set A is also a member of aset B, then A is called a subsetof B

In other words,

if x A x B for all x,

then A is a subsetof B

It is written as AB or BA

A is called a proper subsetof B, if A B and Ais not equal to B.

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In any application of set theory, all the sets are

likely to be subsets of a fixed set. This set is calledthe universalset and is usually denoted by U

Example 1: In studies of rivers, the universal set consist of

all rivers in the world

If two sets have no common element, they are said to be

disjoint

Example 2 : X={1,3,5,7} and Y={2,4,6,8} are disjoint as there is notcommon element

Example 3 : R={1,2,3,4} and S={3,4,5,6} are notdisjoint as there are

two common elements

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Venn Diagram to represent sets

U is the universal set.

A and B are disjoint sets

R is a subset of S

UU

A

B

S

R

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Basic set operations

The unionof sets A and B is the set of allelements which belong to either A or B or both. Itis represented as A U B

The intersectionof sets A and B is the set ofelements which are common to both A and B. Itis represented as A B

Let A={1,2,3,4}, and B={3,4,5,6}then, A U B = {1,2,3,4,5,6},

and A B = {3,4}

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Venn Diagram

The rectangle represents Universal Set. The shadedarea is the union of two sets A and B i.e. A U B

A B

A U B

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Venn Diagram

The rectangle represents Universal Set. Two sets A and

B. The shaded area is their Intersection

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Some more set operations

The differenceof sets A and B is the set ofelements which belong to A but not to B. It

is denoted by A B and read as A minusB

The complementof a set A is the set ofelements which do not belong to A. This isthe difference of the universal set U and AIt is denoted as A/ or Ac

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March 20, 2012Copyright Box Hill Institute

Some examples

Let R={a,b,c,d,e} and S={a,c,e,g,i}

R - S ={b,d}, S - R={g,i}

Let X={0,1,2,3,4,.} and Y={0,2,4,6,}XY = {1,3,5,.}, Y - X =

Let U ={all english alphabets},

V = {a,e,i,o,u},V/= {all consonants}

Note that V/ = U - V

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A few more definitions

The symmetricdifference of two sets Aand B consists of the elements which are

either in A or in B but not in both. It isdenoted as A B

The cardinalityof a set is the number ofelements it contains. It is represented as#A

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Some more examples

Suppose A={1,2,3,4,5} andB={4,5,6,7,8,9,10}

Then, A B = {1,2,3,6,7,8,9,10}

#A = 5, #B = 7, # A B = 8.

Note that # = 0, as it does not containany element.

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Definitions again

Let A and B represent two sets. We havethe definitions in a compact manner

1. A U B ={ x | x A or x B or x both}

2. A B ={ x | x A and x B }

3. A B ={ x | x A and x B }

4. A/={ x | x A }

5. A B={ x | x A or x B but x both}

6. #A = Number of elements in set A

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Some important sets

The set of natural numbers: N = {0,1,2,}

The set of integers: Z = {,2,1,0,1,2,}

The set of positive integers: Z+ = {1,2,3,}

The set of rational numbers Q ,Q = {a/b | a,b Z, b0}

The set of real numbers: R

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Summary

A set is a well-defined list, collection orclass of objects.

There are two forms for presentation of aset : Tabular form and Set builder form.

If every element in a set A is also a

member of a set B, then A is called asubsetof B.

Venn diagrams are used to represent sets.

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References

S Lipschutz : Theory and Problems ofSet Theory and Related Topics,

Schaum's Outline Series, McGraw Hill. http://mathworld.wolfram.com

W1_Sets

Documents

Transcript of W1_Sets