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Chapter Two

Chapter Two. Basic Concepts of Set Theory Symbols and Terminology Venn Diagrams and Subsets. What is a Set?. Set is a collection of Objects. Objects belonging to the set are called elements of the set, or members of the set. Sets are described in three ways. Word descriptions

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Chapter Two

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  1. Chapter Two • Basic Concepts of Set Theory • Symbols and Terminology • Venn Diagrams and Subsets

  2. What is a Set? • Set is a collection of Objects Objects belonging to the set are called elements of the set, or members of the set.

  3. Sets are described in three ways • Word descriptions The set of even counting numbers less than ten • Listing method {2,4,6,8} • Set-builder notation { X| X is an even counting number less than 10}

  4. Suppose E is the name for the set of all letters of the alphabet. Then we can write E = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} • We can shorten a listing by using ellipsis points. For example: E = {a,b,c,d,…,x,y,z}

  5. Examples List the elements of the set of each of the following: • The set of counting number between six and thirteen Answer: { 7, 8, 9, 10,11,12}

  6. B) List each element of the set { 5,6,7,…10} Answer: Completing the list we get {5,6,7,8,9,10} C) {X | X is a counting number between 6 and 7} Answer: There are no elements – so we write { } or 0

  7. Empty or Null Set • Empty set is denoted 0 or { } • Do not use { 0 } or { 0 } to denote the empty set. • Empty set is denoted 0 or { } • Do not use { 0 } or { 0 } to denote the empty set.

  8. Sets of Numbers • Natural or Counting Numbers {1,2,3,4,…} • Whole Numbers {0,1,2,3,4,…} • Integers {…,-1,0,1,….}

  9. Rational Numbers {p/q | p and q are integers and q not equal to 0. (ex. ¾, -7/5, ½ or .55, .67 etc….) • Real Numbers {x | x can be written as a decimal } • Irrational Numbers {x | x is a real number and x cannot be written as a quotient of integers}

  10. Cardinal Numbers • The number of elements in a set is called the cardinal number, or cardinality of the set. The symbol n(A) is read “n of A” and represents the cardinality of set A. • If elements are repeated in a set, they should not be counted more than once when determining the cardinality of the set. For example, if the set, B = { 1,1,2,2,3,3} there are three distinct elements in the set and n(B) = 3

  11. Examples • Find the cardinal number of each of the following sets: • K = {2,4,8,16} n(K) = • M = {0} n(M) = • R = { 4,5,…,12,13} n(R) = • Empty set 0 n(0) =

  12. K = {2,4,8,16} n(K) = 4 • M = {0} n(M) = 1 • R = { 4,5,…,12,13} n(R) = 10 • Empty set 0 n(0) = 0

  13. Finite and Infinite Sets • If the cardinal number of a set is a whole number or a counting number – then that set is finite set. We can count it. • Example: B = { 1,2,3,4,5,6,7,8,9,10} • Some sets are so large we cannot count the elements in the set. • If the set is so large that its cardinal number is not found among the whole numbers, we call that an infinite set. • For example the set of counting numbers is an infinite set. • Example B = {1,2,3,4,….}

  14. Exercise • Review – what are the three common ways to write set notation? • Word Description • Listing Method • Set Builder Notation • Now, write the set of all odd counting numbers using a word description, listing method, and set builder notation

  15. Set Equality • Set A is equal to set B provided the following two conditions are met: • Every element of A is an element of B and • Every element of B is an element of A.

  16. Examples • True or False …. • {a,b,c,d} = {d,c,b,a} • {1,0,1,2,3,} = {0,1,2,3} • {4,3,2,-1} = {3,2,4,1} • {4,3,2,-1} = {3,2,4,1} True True False True

  17. Venn Diagrams and Subsets • Universe of Discourse • For a problem includes all things under discussion at a given time. Suppose the NOVA Loudon campus considered raising the scores for the Algebra 1 placement exam. The universe of discourse might be all potential students wishing to take Algebra 1 from the Loudon campus.

  18. Universal Set • In mathematical theory of sets, the universe of discourse is known as the Universal Set. • The letter U is usually used for the universal set.

  19. Venn Diagrams • The universal set is represented by a rectangle, and other sets of interest within the universal set are represented by an oval region, circles, or other shapes.

  20. U Venn Diagrams The entire region bounded by the rectangle represents the Universal Set - U A’ The oval represents the Set A A The region inside U and outside the oval is labeled A’ (read A prime) This is the compliment of A Contains elements in U not in A

  21. Compliment of a Set • For any set A within the universal set U, the complement of A, written A’ is the set of elements of U that are not elements of A . A’ = { X | X E U and X E A}

  22. Subset of a Set • How do we define the compliment of the universal set, U’. • The set U’ is found by selecting all the elements of U that do not belong to U. A U

  23. A • For the universal set U • U’ = 0 • Next, lets look at the compliment of the empty set, 0’. • Since 0’ = { X | X E U and X E 0 } and set 0 contains no elements, every member of the universal set U satisfies this definition U

  24. So, for every universal set U, 0’ =U • Suppose U = {1,2,3,4,5} • Let A = {1,2,3} • Every element of A is an element of set U • Set A is called a subset of set U

  25. Subset of a Set • Set A is a subset of set B if every element of A is also an element of B. • AB Examples

  26. Set Equality • If A and B are sets, then A = B if • A B and B A. Suppose B = { 5,6,7,8} and A= {6,7}. The A is a subset of B, but A is not all of B. A is called a proper subset of B. A B.

  27. Proper Subset of a Set • Set A is a proper subset of set B if A B and A = B Then A B.

  28. Set A is a subset of set B, if every element of set A is also an element of set B. • Set A is a subset of Set B, if there are no elements of A that are not also element of B • IS the empty set a subset of A or B or both? • 0 B • The empty set is a proper subset of every set except itself • Every set (except the empty set) has at least two subsets, the set itself and the empty set.

  29. Finding the number of subsets • Number of Subsets • The number of subset of a set with n elements is 2 n • Since the value 2 n contains the set itself, we must subtract 1 from this value to obtain the number of proper subsets of a set containing n elements. • The number of proper Subset of a set with n elements is 2n -1

  30. Homework • Exercises • 2.1 Page 54 9 -21 odd, 25, 27, 29, 31, 33, 35, 41 -49 odd, 59 – 66 odd, 67 – 78 odd 2.2 Page 61 7 -41 odd, 43, 45, 49, 52

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