Math10 1 Lecture1

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First Lecture on College Algebra

Math10 1 Lecture1

  1. 1. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM A SET is a well-defined collection of objects. <ul><li>Examples: </li></ul><ul><li>Set of books found in Mapua Library Makati </li></ul><ul><li>Set of players of Spain’s 2010 soccer team </li></ul><ul><li>3. Set containing all the months in a year </li></ul><ul><li>4. Set of students enrolled in Math 10-1 AY01 class for 1 st term AY 2010-2011 </li></ul>
  2. 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. 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. 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:
  5. 5. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM <ul><li>Counting numbers which are multiples of 3 and less than 20. </li></ul><ul><li>Single digits used in our decimal system. </li></ul><ul><li>3. Set of all odd numbers between 2 and 12. </li></ul>Illustration:Write each of the following using roster and rule method:
  6. 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. 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. 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. 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. 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. 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. 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. 13. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM Venn Diagram A B B A U U
  14. 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. 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
  16. 16. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM <ul><li>Given the sets A = {1,2,3}, B = {0,1,2,3,4}, </li></ul><ul><li>C = {1,3,5,8} and D = {5,10,15}, find </li></ul><ul><li>A  B 3. A  B </li></ul><ul><li>2. C  D 4. C  D </li></ul>Illustrations:
  17. 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. 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
  19. 19. Math 10-1: College Algebra SETS AND THE REAL NUMBER SYSTEM <ul><li>Given the sets A = {1,2,3}, B = {1,3,5,8}, </li></ul><ul><li>and U = {1,2,3,…,9,10}, find </li></ul><ul><li>A - B 3. A’ </li></ul><ul><li>2. B - A 4. B’ </li></ul>Illustrations:
  20. 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. 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:

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