Math10 1 Lecture1

Math 10-1: College Algebra

SETS AND THE REAL NUMBER SYSTEM

A SET is a well-defined collection of objects.

Examples:1.Set of books found in Mapua Library Makati2.Set of players of Spain’s 2010 soccer team 3. Set containing all the months in a year4. Set of students enrolled in Math 10-1 AY01 class for 1st term AY 2010-2011



The cardinality of set A is the number of elements contained in A and is denoted by |A|.

Determine the cardinality of the previously given sets.



Two ways of writing a set:

Rule Method: >>describes a set by some ruleRoster Method:>>list down all the elements of the set.



Rule Method: {x|x is a positive integer less than 6}

Roster Method:{1,2,3,4,5}

Illustration:



1. Counting numbers which are multiples of 3 and less than 20.

2. Single digits used in our decimal system.3. Set of all odd numbers between 2 and

12.

Illustration:Write each of the following using roster and rule method:



>The symbol { } denote the set that is empty.>The symbol є literally means ‘is an element of’ or ‘belongs to’>The symbol U denotes the universal set, set containing all elements in consideration.

Some notations:



A one-to-one correspondence exists between two sets A and B if it is possible to associate the elements of A with the elements of B in such a way that each element of each set is associated with exactly one element of the other.

SET RELATIONSHIPS:



>Two sets A and B are equivalent, denoted by AB, if and only if there exists a one-to-one correspondence between them.>Two sets A and B are equal, denoted by A = B, if the elements of A and B are exactly the same.

Equal and Equivalent Sets



Two sets A and B are joint sets if and only if A and B have common elements; otherwise, A and B are disjoint.

Joint and Disjoint Sets



>A set A is a subset of B, A B, if every element of A is in B.

>If for A B, B contains elements that are not in A, then A B. (proper subset)

Subset and Proper Set



The power set of A is the set containing all subsets of A and is denoted by (A).

Power Set



Venn Diagram is the pictorial representation of sets and (is usually symbolized by circles and rectangles.)

Venn Diagram



Venn Diagram

U UAB

BA



Union of SetsThe union of two sets A and B, denoted by A B, is the set whose elements belong to A or B or both to A and B. In symbol, A B = {x|x A or x B or x A and B}

Operations on Sets



Intersection of SetsThe intersection of two sets A and B, denoted by A B, is the set whose elements are common to A and B. In symbol, A B = {x|x A and x B}

Operations on Sets



Given the sets A = {1,2,3}, B = {0,1,2,3,4},C = {1,3,5,8} and D = {5,10,15}, find1.A B 3. A B2. C D 4. C D

Illustrations:



Difference of Sets,The difference of two sets A and B, denoted by A - B, is the set whose elements are in A but not in B. In symbol, A - B = {x|x A and x B}

Operations on Sets



Complement of a Set,The complement of a set A, denoted by A’, is the set with elements found in the universal set U, but not in A, i.e., the difference of the universal set U and A. In symbol, A’ = {x|x U and x A} = U - A

Operations on Sets



Given the sets A = {1,2,3}, B = {1,3,5,8},and U = {1,2,3,…,9,10}, find1.A - B 3. A’2. B - A 4. B’

Illustrations:



Cartesian Product,The Cartesian product of two sets A and B denoted by AxB, is the set of ordered pairs(x,y) suct that x is an element of A and y is an element of B. In symbol, AxB = {(x,y)|x A and y B}Note: A x B B x A

Operations on Sets



Suppose a particular menu in a burger joint includes the following:Hamburger (b) Soda (s)Cheeseburger (c) Tea (t)Hotdog sandwich (d) Fruit Juice (f)

What are the possible combinations of burger and drinks?

Illustration:

Math10 1 Lecture1

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