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e LEARNING - MATHEMATICS:

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# TOPIC: SET THEORY

WHAT IS A SET IN MATHEMATICS

A set is a collection of distinct objects. These objects must be well defined and must have something in common. It could be a group of numbers, persons etc. For instance, the students of a particular class or taking a particular subject or writing a particular exam can form a set. Also, a list of even numbers between 0 and 50 can form a set. We could have a set of integers between 1 and 20.

There are two ways of writing a set: (a) Enumeration, and (b) Description.
(a) Enumeration method: When we use enumeration to write a set, we will write out all the individual elements of the set. For instance, if set T is all real numbers from 1 to 5, then, we can write out the elements of the set thus: T = {1, 2, 3, 4, 5}.

(b) Description method: This is the use of statement or symbol to describe the elements of a set. For instance, if set S is all positive integers, we will write the set as: S = {x | x a positive integer} which reads: "S is the set of all x-numbers, such that x is a positive integer". Note that a vertical bar (or use colon) is inserted to separate the general symbol for the elements from the description of the elements. Using another example, if set M is all real numbers greater than 2 but less than 6; we can express this symbolically thus: M = {x : 2 < x < 6}.

The objects in a set are called the elements of the set and they are indicated by the symbol , which is read as: "is an element of". So if a set A is defined as: A = {2, 4,6, 8}. The elements of the set are 2, 4, 6, 8. Therefore 2 A, meaning 2 is an element of A. Also, 4 A, 6 A, and 8 A.

However, if an object is not an element of a set, it is indicated by the symbol , which is read as: "is not an element of".So if a set A is defined as: A = {2, 4, 6, 8}. It is obvious that 1, 3, 5 are not elements of the set. Therefore 1 A, meaning 1 is not an element of A. Also, 3 A, and 5 A.

The number of elements in a set is called the cardinality of the set and it is indicated by the symbol "n". In the case of set A = {2, 4, 6, 8}. The set contains 4 elements, therefore the cardinality of A = 4 i.e n{A} = 4. Using another example, if a set B = {1, 3, 5, 7, 9}; n{B} = 5. If set J = {2, 5, a, c, 7, e, 8, f}; n{B} = 8.

A set that contains a finite or countable number of elements is called finite set; for example A = {2, 4, 6, 8} contains four (4) elements which is countable. A set that contains an infinite or uncountable number of elements is called infinite set; for example S = {x | x a positive integer} contains all positive integers i.e 1, 2, 3, 4 etc which is uncountable. A set that contains no element is called an empty or null set and it is represented by the symbol or . Note that it is not the same as because there is an element in the latter while does not contain any element. So, is not an empty set because it contains as element.

RELATIONSHIP BETWEEN SETS
There are many possible kinds of relationship that can exist between two sets. Lets examine some of them:

(A) Equal Sets: Two or more sets are said to be equal if they have the same elements irrespective of their arrangement. For example, if set A = {2, 4, g, h, 6, 8}, and set B = {6, h, 4, 8, g, 2}. Since the same elements that are in set A are also in set B then, A = B.

(B) Subset of a set: A set is a subset of another set if all its elements are contained in the other set. For example, if set M = {3, 6, 1, 5, 4} and set N = {1, 3}. It is obvious that all the elements of set N are in set M. So N is a subset of M. The symbols of subset are: (is contained in) and (includes); so we can write N M (N is contained in M) or we write M N (M includes N). Also note that the symbol means not a subset of or not contained in, and the symbol means not include. If P = {2, 9, 8}, it is obvious that the elements of set P are not in set M, so we can write that P M (P is not a subset of or not contained in M) or M P (M does not include P). Also, if Q = {7, 6, 5, 2}, you will observe that some elements of Q are contained in M (e.g. 6, 4) but some elements are not (e.g. 7, 2). Since not all the elements of Q are in M then, Q M (Q is not a subset of or not contained in M).

It is important to note that the subsets of a set include the set itself and also an empty set as well as all the possible combination of the elements. The total number of subsets that can be obtained from a set can be known using the formular where n is the number of elements in the set. Lets take some examples to illustrate this.

Question: Given that A = {2, 5}, write out all the subsets of A.

Answer: The first thing to do is to know the total number of subsets that can be obtained from set A by using the formular . Also, you must remember that the subsets of a set includes the set itself and an empty set.
Total number of subset = where n is the number of elements in the set.
Since Set A has 2 elements, then, the total number of subsets for set A = = 4.

The subset of A = {2, 5} are: first {2}, second {5}, third {2, 5}, and fourth { }.

Question: Given that Z = {b, f, s}, write out all the subsets of Z.

Answer: The total number of subsets that can be obtained from set Z can be known using the formular where n is the number of elements in the set. Also, the subsets of all sets include the set itself and an empty set.

Since Set Z has 3 elements, then, the total number of subsets for set Z = = 8.

The subsets of set Z are: {b}, {f}, {s}, {b, f}, {b, s}, {f, s}, {b, f, s}, and { }. 