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, $W=\{x:xisadayoftheweek\}$ or $Y=\{x|xisaprimenumberand0<x<24\}$

2. The rule method: $W=\left\{daysoftheweek\right\}$

3. 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 $set1$ has one and only one partner in $set2$. 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 $F$ ($D\not\subset F$) and $F$ is not a superset of $D$ ($F\not\supset D$).

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 $\epsilon$. Considering the number of students in all the departments and faculties in the University of Port Harcourt, the universal set can be represented as $\epsilon =\{studentsintheUniversityofPortHarcourt\}$.

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 $\{a,b,c\}$.