1.6 Subsets and Superset

Subsets: A set A is said to be a subset of set B if every element of \(\mathrm{A}\) is also an element of \(\mathrm{B}\). In symbols we write \(\mathrm{A} \subset \mathrm{B}\), if \(a \in \mathrm{A} \Rightarrow a \in \mathrm{B}\).

Example 1: \(A=\{1,2,3\}, \quad B=\{1,2,3,4\}\), In this example, all the elements of set A are in Set B. Therefore we can say set A is a subset of B. \(\mathrm{A} \subset \mathrm{B}\) In this case, we can not say B is a subset of A because all the elements in set B is not present in set A. In this case, we say B is a superset of A.

Note: An important point to remember is that any set is a subset of itself. Also null set is a subset of every set.

We denote set of real numbers by \(\mathbf{R}:\{\pi,1,3\cdots\}\)
set of natural numbers by \(\mathbf{N}:\{1,2,3,4,5 \ldots\}\) 
set of integers by \(\mathbf{Z}:\{-7,-6,-5,1,4,5, . .\}\) 
set of rational numbers by \(\mathbf{Q}:\{1.2,1.3,1.5,2.2 \ldots\}\) 
set of irrational numbers by \(\mathbf{T}:\{\pi=3 \cdot 14159265 \ldots, \sqrt{2}=1 \cdot 414213 \cdots\}\)

We observe that,

\(\begin{aligned}&\mathbf{N} \subset \mathbf{Z} \subset \mathbf{Q} \subset \mathbf{R} \\
&\mathbf{T} \subset \mathbf{R}, \mathbf{Q} \not \subset \mathbf{T}, \mathbf{N} \not \subset \mathbf{T}\end{aligned}\)

Let us consider few examples

(i) Let \(\mathrm{A}=\{1,3,5\}\) and \(\mathrm{B}=\{x: x\) is an odd natural number less than 6\(\}\). Then \(\mathrm{A} \subset \mathrm{B}\) and \(\mathrm{B} \subset \mathrm{A}\) and hence \(\mathrm{A}=\mathrm{B}\).

(ii) Let \(\mathrm{A}=\{a, e, i, o, u\}\) and \(\mathrm{B}=\{a, b, c, d\}\). Then \(\mathrm{A}\) is not a subset of \(\mathrm{B}\), also \(\mathrm{B}\) is not a subset of \(\mathrm{A}\).

(iii) Let \(\mathrm{A}\) and \(\mathrm{B}\) be two sets. If \(\mathrm{A} \subset \mathrm{B}\) and \(\mathrm{A} \neq \mathrm{B}\), then \(\mathrm{A}\) is called a proper subset of \(\mathrm{B}\) and \(\mathrm{B}\) is called superset of \(\mathrm{A}\). For example, \(\mathrm{A}=\{1,2,3\}\) is a proper subset of \(\mathrm{B}=\{1,2,3,4\}\). If a set \(\mathrm{A}\) has only one element, we call it a singleton set. Thus, \(\{a\}\) is a singleton set.

Example 2: Consider the sets
\(
\phi, \mathrm{A}=\{1,3\}, \quad \mathrm{B}=\{1,5,9\}, \quad \mathrm{C}=\{1,3,5,7,9\} \text {. }
\)
Insert the symbol \(\subset\) or \(\not \subset\) between each of the following pair of sets:
(i) \(\phi \ldots B\)
(ii) A… B
(iii) A…C
(iv) B …C

Solution:

(i) \(\phi \subset \mathrm{B}\) as \(\phi\) is a subset of every set.
(ii) \(\mathrm{A} \not \subset \mathrm{B}\) as \(3 \in \mathrm{A}\) and \(3 \notin \mathrm{B}\)
(iii) \(\mathrm{A} \subset \mathrm{C}\) as \(1,3 \in \mathrm{A}\) also belongs to \(\mathrm{C}\)
(iv) \(\mathrm{B} \subset \mathrm{C}\) as each element of \(\mathrm{B}\) is also an element of \(\mathrm{C}\).

Example 3: Let \(A, B\) and \(C\) be three sets. If \(A \in B\) and \(B \subset C\), is it true that \(\mathrm{A} \subset \mathrm{C}\)?. If not, give an example.

Solution: No. Let \(\mathrm{A}=\{1\}, \mathrm{B}=\{\{1\}, 2\}\) and \(\mathrm{C}=\{\{1\}, 2,3\}\). Here \(\mathrm{A} \in \mathrm{B}\) as \(\mathrm{A}=\{1\}\) and \(\mathrm{B} \subset \mathrm{C}\).

But \(\mathrm{A} \not \subset \mathrm{C}\) as \(1 \in \mathrm{A}\) and \(1 \notin \mathrm{C}\).
Note that an element of a set can never be a subset of itself.

Intervals as subsets of \(R\):

Open interval and closed interval are used to represent a range of numeric values. The open interval includes only the values between the endpoints and is represented as ( ). The closed interval includes even the endpoints of the range of values and is represented as [ ]. The open interval can be presented as an expression \(a<x<b\), and the closed interval is represented as an expression \(a \leq x \leq b\).

Open interval

Open interval is the one in which endpoints are not included. The numbers are written as order pair between plain bracket i.e open interval a to \(b\) open is written as \((a, b)\). Every real number from \(a\) to \(b\) is included except \(a\) and \(b\) itself. We get open interval when we use inequality like

• \(<\) less than
• \(>\) greater than

Example 4: All values of \(x\) greater than \(-3\) and less than 5 are between \(-3\) and 5 but \(-3\) and 5 are not included \(x \in(-3,5)\). We can write it as

\(-3<x<5 \text { Or } x \in(-3,5)\)

Closed interval

Closed interval is the one in which endpoints are included. The numbers are written as pair between square bracket i.e Closed from a to b and is written as [a, b]. Every real number from a to \(b\) including \(a\) and \(b\) are included. Closed interval is used while writing inequalities

• \(\leq\) less than or equal to
• \(\geq\) greater than or equal to.

Example 5:  All values of \(x\) greater than or equal to \(-3\) and less than or equal to 5. In this both \(-3\) and 5 are included, is written as \(x \in[-3,5]\). We can write it as 

\(-3 \leq x \leq 5 \quad \text { Or } \quad x \in[-3,5]\)

Example 6: The expression \(2<x<6\) (does not include the end points 2 and 6), represents an open interval, and the expression \(\mathrm{2} \leq \mathrm{x} \leq \mathrm{6}\) (includes the end points 2 and 6) represents a closed interval.

Let \(a, b \in \mathrm{R}\) and \(a<b\). Then
(a) An open interval denoted by \((a, b)\) is the set of real numbers \(\{x: a<x<b\}\). All the points between \(a\) and \(b\) belong to the open interval \((a, b)\) but \(a, b\) themselves do not belong to this interval.
(b) A closed interval denoted by \([a, b]\) is the set of real numbers \(\{x: a \leq x \leq b)\)
(c) Intervals closed at one end and open at the other are given by
\(
\begin{aligned}
&{[a, b)=\{x: a \leq x<b\}} \text { is an open interval from } a \text { to } b \text {, including } a \text { but excluding } b \text {. }\\
&(a, b]=\{x: a<x \leq b\} \text { is an open interval from } a \text { to } b \text { including } b \text { but excluding } a \text {. }
\end{aligned}
\)

On real number line, various types of intervals described above as subsets of R, are shown in the figure below.

Example 7: Write as intervals :\(\{x: x \in R,-4<x \leq 6\}\)

Solution: \((-4,6]\) (-4 not included, so it should be open and 6 included, so it should be closed)