Introduction, Definition and Set Notation - SS1 Mathematics Lesson Note

A set is a collection of objects according to a well-defined property. Said objects are called the elements or members of the set. Sets are usually denoted with uppercase letters \(A,\ B,\ C\), and so on. There are three basic ways of describing sets:

1. The set builder or property method: This describes the elements by referring to their common property. Example, \(W\ = \ \{ x:\ x\ is\ a\ day\ of\ the\ week\}\) or \(Y\ = \ \{ x|\ x\ is\ a\ prime\ number\ and\ 0 < x < 24\}\)

The rule method: \(W = \left\{ days\ of\ the\ week \right\}\)

The listing or roster or tabular method: All the elements of the set are listed\(.\ Y\ = \ \{ 2,3,5,7,11,13,17,19,23\}\)

From, the set of all prime numbers between \(0\) and \(24\), that is \(Y\ = \ \{ 2,3,5,7,11,13,17,19,23\}\), \(11\) is an element of \(Y\) but \(15\) is not. We state that \(11\) is an element of \(Y\) mathematically as \(11 \in \ Y\) and that \(15\) is not a member of \(Y\) as \(15\ \notin \ Y\).

Finite and Infinite Sets

If the elements of a set have a definite number such as the days of the week, such set is termed a finite set. Otherwise, it is an infinite set such as the set of all natural numbers.

Equivalence and Equality of Sets

Two sets \(A\) and \(B\) are said to be equivalent if they both contain the same number of elements. Each member in \(set\ 1\) has one and only one partner in \(set\ 2\). For instance, \(A\ = \ \{ 2,3,6,8\}\) and \(B\ = \ \{ a,b,c,d\}\) are equivalent to each other since both sets have 4 elements each, that is, \(A \equiv B\).

Two sets \(M\) and N are said to be equal if their elements are the same irrespective of arrangement. For instance, \(M\ = \ \{ 2,3,6,8\}\) and \(N\ = \ \{ 3,2,2,6,8,8,3\}\) are equal since both sets have the elements \(2,\ 3,\ 6\) and \(8\), that is \(M = N\). However, they are not equivalent as \(M\) has \(4\) members and \(N\),\(\ 7\)

Sets \(X\ = \ \{ 5,3,4,2\}\) and \(Y\ = \ \{ 2,3,4,5\}\), are both equal and equivalent.

Null or Empty Sets

A set that has no members is a null or empty set. It is denoted mathematically as \(\varnothing\) or \(\{\}\). Example, the set of Senior Secondary School students under the age of 6.

Subsets and Supersets

Given two sets, \(A\) and \(B\), \(A\) is said to be a subset of \(B\) if and only if all the elements of \(A\) are contained in \(B\). This is denoted \(A\ \subset B\). If \(A\) is a subset of \(B\) then \(B\) is a superset of \(A\) because \(B\) contains all the members of \(A\), that is \(B\ \supset A\).

Given \(D = \{ 2,3,5\}\) and \(E = \{ 2,4,6,7,5,7,8,2,3\}\), \(D\) is a subset of \(E\) (\(D \subset E\)) and \(E\) is a superset of \(D\) (\(E \supset D\)).

Given \(D = \{ 2,3,5\}\) and \(F = \{ 11,2,3,17,19\}\), \(D\) is not a subset of \(F\) (\(D ⊄ F\)) and \(F\) is not a superset of \(D\) (\(F ⊅ D\)).

NOTE: Interesting, the null set \(\{\}\) or \(\varnothing\) is a subset of every other set and every set is a subset of itself.

Universal set

In a particular context, a universal set is a set of all elements under consideration. It is denoted with \(U\) or \(\varepsilon\). Considering the number of students in all the departments and faculties in the University of Port Harcourt, the universal set can be represented as \(\varepsilon = \{ students\ in\ the\ University\ of\ Port\ Harcourt\}\).

Power Set

The Power set of a set \(A\) containing \(n\) members is number of possible subsets of \(A\) and is denoted as \(P(A) = \ 2^{n}\). A set of only \(3\) elements, \(X = \{ a,b,c\}\) where \(n = 3\), will have a power set of \(2^{n} = 2^{3} = 8\), meaning \(X = \{ a,b,c\}\) has 8 possible subsets as follows: \(\{\}\), \(\{ a\}\), \(\{ b\}\), \(\{ c\}\), \(\{ a,b\}\), \(\{ a,c\}\), \(\{ b,c\}\) and \(\left\{ a,b,c \right\}\).