Sets and Methods of describing sets

A set can be defined as a collection of objects according to a well defined common elements or property. The main purpose of this their common property is foe easy identification. For instance, we hear of under 13 football players, meaning “the set of football players” whose ages fall below 13 years; “set of school uniform {of the dresses and sandals.

Notation and Methods of Describing Sets

A set is usually represented by a capital letter, for example,

A = The set of even numbers less than 110

B =  The set of Nigerian Presidents since Independence in 1960

C =  The set of months of the year

There are basically three ways of representing the sets, namely;

1. The set builderor property form: This describes the elements of the set by referring to their common property. This method of describing a set is called the set-builder method.

e.g. W = {x : x is the day of the week}

Y = {even numbers between 0 and 10}

or     Y = {x : x is even number and 0 < x < 19}

2. Also the above example can be given by the rule methodas W = {the days of the week}

This is using the rule method since any day of the week is in set W

3. The roster of tabular or listing method: this method actually lists all the members of the set

e.g.   w = {Sunday, Tuesday, Wednesday, Thursday, Friday, Saturday}

Y = {2,4,6,8}

Note: The elements in 3 above and the descriptions in 1 and 2 are usually enclosed in curly brackets or braces as in the examples above. In the case of roster or tabular listing form, the elements are separated by commas as shown above.

Using the example above, Monday “is a member of” W. This is denoted as Monday Î W read as

Monday “belongs to” W or Monday “is an element of” W or Monday “is in” W. If a member “does not belong to” the set of days in the week, this can be denoted by Ada Ï W read as ‘Ada is not in’ W or Ada “is not an element of” Wor Ada “is not a member of” W. Generally, the elements or members of set are usually the lower case letters such as a, b, c, d, …. , while the whole set itself is usually represented with a capital letter A, B, W, Y, ……

Finite and Infinite Set

As already mentioned above, some collections of objects according to a well defined common property can be large or small and their members or elements definite or infinite. If the members of a set have a definite numbers like the days of the week, we term this as finite, otherwise it is infinite like the set of natural numbers or counting numbers.

Examples of finite and infinite sets are as follows:

1. W = {days in the week} – W is finite
2. N = {x | x is a multiple of 2} – N is infinite

These can be written in tabular form as

W = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} which contains seven elements and therefore is finite.

N = {2,4,6,8, ….} which contains indefinite number elements and therefore is infinite.

Others examples are

1. M = {months in the year} – M is finite
2. F = {y | y is odd} – F is infinite.

Null Set or Empty Set

Wherefore there is no body in the classroom (like after school hours or during the holidays), the classroom is empty of persons at that time. We refer to it as an empty or null set. A null set is therefore defined as a set that contains no elements. This set is denoted by { } or . (Note that when this  is used it is not enclosed in a bracket. Hence the set {.} or {} is not an empty set rather it is a set containing an element “.” Or “”).

Example of a Null Set

1. The set of students in Nigeria secondary schools below 5 years is empty. That is A = {Students in secondary schools in Nigeria below 5 years} = { } or because no student below five years in Nigerian secondary school.
2. B = {x | x²= 4, x odd}. \the solution is 2 and -2 neither of which is an odd number, so B = { }.

iii. C = {women priests in Roman Catholic Church} is empty because there is no woman priest in Roman Catholic Church, so C = { } or .

Subsets and Supersets

Subsets are sets which are contained in another set or given two sets A and B, set A is said to be a subset of a set B if and only if all the elements of set A are contained ion set B. This is denoted by A  B.

Examples

1. If two sets D = {0, 2, 4, 6, 8} and E = {2, 4, 6}, the E is a subset of D because all the elements of E are in D.
2. If A = {Garri, cassava,yam , beans}, B = {Garri, beans} then B is a subset of A. Hence if A set P is contained another set Q, the relationship is denoted by “P Q” which is read as “P is contained in Q” or “P is a subset of Q”.

iii. X = {a, b, c, d, e} and Y = {a, b, c}. Y is a subset of X

1. If F = {x | x is a positive integer}, F = {1, 2, 3, 4, 5, 6,….}, G = {y | y is a multiple of 3} i.e. G = {3, 6, 9, 12….} then G is contained in F i.e. G F.
2. Since the null or empty set contains no elements, it can be contained in any set without changing the set. Hence the null set or empty is a subset of every set.
3. Every set is a subset of itself because all the elements in the set are contained in the set itself. These two sets, the null set or the set itself are called improper subsets because they are special case s of subsets while all the others are called proper subset.

Universal Set

The set under consideration are likely to be the subsets of a fixed or a global set. This fixed or global set is called the Universal Set.

A universal set is the set of all objects of interest in a particular discussion. This set can be very big or small depending on the context.

For example if we are discussing medical students at Nnamdi Azikwe University, the whole students of the Nnamdi Azikwe University can be our universal set. If our reference is to persons, every person in the Nnamdi Azikwe University can be the universal set. Therefore, the universal set varies as our group of reference varies. The universal set is usually denoted by U or e, so

U = {students in Nnamdi Azikwe University} or {every person in Nnamdi Azikwe University}

A = {a, b, c, d} can have a universal set of all letters of the English alphabet, i.e. U = {letters of the English alphabets}; or the sets of {first 13 letters of the English alphabet}; or the {set of first 5 letters of the English alphabets}. This implies that the universal set may not necessarily be unique.

Example

If set U = {1, 2, 3, 4,….}, describes the following sets by the listing method.

1. A = {x | x²– 3x + 2 = 0} b. C = {x | x = 2 or x = 3 or x = 20}

Solution

A = {x | x² – 3x + 2 = 0} here x² – 3x + 2 = 0 can be factorized into (x -2)(x -1) = 0, hence x = 2 or 1 so A = {1, 2}.

20. C = {2, 3, 10}. This kind of question requires nothing other than listing the solutions in the set 2, 3,20.