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Math102 Fall 2004 Homepage mathdstate

2.1 Basic Properties of Sets. A Set may be thought of as a collection of objects. A Set can be presented as elements, members, it is called the roster form.A = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } as the set of all the single digits.B = { a, e, i, o, u } as the set of vowels in the English alphabet.

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Math102 Fall 2004 Homepage mathdstate

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    1. Math102 Fall 2004 Homepage http://math.indstate.edu/chi/ma102

    2. 2.1 Basic Properties of Sets A Set may be thought of as a collection of objects. A Set can be presented as elements, members, it is called the roster form. A = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } as the set of all the single digits. B = { a, e, i, o, u } as the set of vowels in the English alphabet.  n(A) the cardinal number of set A

    3. Properties of Set N = { 1, 2, 3, 4, 5, ... } Counting numbers (or Natural numbers) W = { 0, 1, 2, 3, 4, 5, ... } Whole numbers (or non-negative integer) I = { ...., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, .... } Integer Q={all terminating, non-repeating decimals} Rational Numbers X={all non-terminating, non-repeating decimals} Irrational Numbers R={all rational or irrational numbers} Real Numbers

    4. Equal Sets and Equivalent Sets Equal sets means two sets that contain exactly the same elements Two sets are equivalent if they have the same number of elements

    5. EXAMPLE 1. Use the roster method to write each of the given sets. a. The set of natural numbers less than 5 b. The solution set of x + 5 = -1 c. The set of negative integers greater than -4

    6. Solution::: a. The set of natural numbers or counting numbers is given by {I, 2, 3, 4, 5, 6, 7, .. .}. The natural numbers less than 5 are 1,2,3, and 4. Using the roster method we write this set as {1, 2, 3, 4}. b. Adding -5 to each side of the equation produces x = -6. The solution set of x + 5 = -1 is {-6}. c. The set of negative integers greater than -4 is {-3, -2, -1}.

    7. CHECK YOUR PROGRESS Use the roster method to write each of the a. The set of whole numbers less than 4 b. The set of counting numbers larger than 11 and less than or equal to 19 c. The set of negative integers between -5 and 7

    8. Empty Set (Page 52) The empty set, or null set, is the set that contains no elements. The symbol Ř or { } is used to represent the empty set. As an example of the empty set, consider the set of natural numbers that are negative integers.

    9. Set-builder Notation (Page 52) Another method of representing a set is set-builder notation. Set-builder notation is especially useful when describing infinite sets. For instance, in set-builder notation, the set of natural numbers greater than 7 is written as follows. {x | x > 7 and x in an natural number}

    10. EXAMPLE 3 Use set-builder notation to write the following sets. a. The set of integers greater than -3. b. The set of whole numbers less than 1000.

    11. Solution::: a. {x | x is an integer and x > -3} b. {x | x is a whole number and x < 1000}

    12. CHECK YOUR PROGRESS Use set-builder notation to write the following sets. a. The set of integers less than 9 b. The set of natural numbers greater than 4

    13. EXAMPLE Find the cardinality of each of the following sets. a. J = {2, 5, 2, 5, 2} b. S = {3, 4, 5, 6, 7, ..., 31} c. T = {3, 7, 51}

    14. Solution::: a. Set J contains exactly two elements, so J has a cardinality of 2. Using mathematical notation we state this as n(J) = 2. b. Only a few elements are actually listed. The number of natural numbers from 1 to 31 is 31. If we omit the numbers 1 and 2, then the number of natural num­bers from 3 to 31 must be 31 - 2 = 29. Thus n(S) = 29. c. n(T) = 3.

    15. CHECK YOUR PROGRESS Find the cardinality of the following sets. a. C = { -1,5,4,11,13 } b. D = { Ř }

    16. EXAMPLE State whether each of the following pairs of sets are equal, equivalent, both, or neither. a. {a, e, i, o, u}, {3, 6,11,15, 19} b. {4, -2, 7}, {-2, 4, 7, 4}

    17. Solution::: a. The sets are not equal. However, each set has exactly five elements, so the sets are equivalent. b. The first set has three elements and the second set has exactly the same three elements, so the sets are equal and equivalent.

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