Furthermore we can also describe other sets within a particular bigger set like

Furthermore we can also describe other sets within a

This preview shows page 33 - 36 out of 151 pages.

Limpopo for year 2017 to form a set. Furthermore we can also describe other sets within a particular bigger set like for instance taking a set of students in faculty of Science and Agriculture, School of Mathematical and Computer Sciences, Students studying SMTB000, etc. It is possible that certain individuals (or objects) may appear in more than one set. Definition 2.1.1. A set is an unordered collection of well defined objects called the elements or members of the set. “Member of” or “element of” is a relation between objects and sets. If a is an 30
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object and A is a set, then a A means a is a member of A or ‘a is an element of A ’. Then a / A means a is not a member of A or ‘a is not an element of A 2.1.1 Listing of elements There are two methods of representing a set which are: 1. Listing the elements in a roster: Here we list elements between the two curly brackets (braces). e.g. The notation { a, 1 , e, 7 } represents the set with four elements a, 1 , e, and 7. We call this a roster method . Sometimes the roster method can be used to describe a set without listing all the members. This is done by listing some of the members, and then ellipses ( . . . ) are used when the general pattern of the elements is obvious. e.g. D = { 1 , 4 , 9 , 16 , . . . , 81 , 100 } which is clearly a set of all perfect square numbers from 1 to 100. One can easily make note of obvious elements not listed, for example, 36 D and 40 / D . 2. Set builder notation: We characterized the elements of that set by a specific property or properties they must have in order to be a member of such set. e.g. A = { x : x is a prime number between 6 and 15 } . We often use this method when it is impossible to list such elements, though in some instances it can be easy to list such element. From our ex- ample it easy to list elements in set A , i.e. A = { 7 , 11 , 13 } ; but it is not pos- sible to list elements of set B = { x : x = a b , where a and b are negative integers } , since there are infinite such number in set B . 2.1.2 Important sets of numbers: 1. Natural numbers: Natural numbers are also known as counting num- bers, beginning at 1. The set of natural numbers is denoted by N ; i.e. N = { 1 , 2 , 3 , 4 , . . . } . 31
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N is an infinite set. If we include a zero we obtain the set of whole numbers denoted by N 0 . So N 0 = { 0 , 1 , 2 , 3 , 4 , . . . } 2. Integers: Integers consist of natural numbers, negative natural numbers and a zero. The set if integers is denoted by Z ; i.e. Z = { . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . } . Positive integers are integers greater than zero and represented as follows; Z + = { 1 , 2 , 3 , 4 , . . . } . Negative integers are integers less than zero and represented as follows; Z - = {- 1 , - 2 , - 3 , - 4 , . . . } . 3. Rational numbers are numbers that can be written in the form a b where a, b Z and b 6 = 0; i.e. Q = { a b : a, b Z and b 6 = 0 } . Note that all integers are rational numbers with b = 1. 4. Irrational numbers are the opposite of rational, i.e. those numbers that cannot be written as a b where a, b Z and b 6 = 0. These can be represented as R \ Q or R - Q or Q c (complement of Q ), where R is the set of real numbers to be discussed next.
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