First Lecture on College Algebra

2. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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.

3. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Two ways of writing a set: Rule Method : >>describes a set by some rule Roster Method : >>list down all the elements of the set.

4. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Rule Method : {x|x is a positive integer less than 6} Roster Method : {1,2,3,4,5} Illustration:

6. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM >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:

7. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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:

8. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM >Two sets A and B are equivalent , denoted by A  B, 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

9. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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

10. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM >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

11. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM The power set of A is the set containing all subsets of A and is denoted by  (A). Power Set

12. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Venn Diagram is the pictorial representation of sets and (is usually symbolized by circles and rectangles.) Venn Diagram

13. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Venn Diagram A B B A U U

14. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Union of Sets The 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

15. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Intersection of Sets The 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

17. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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

18. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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

20. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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

21. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM 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: