Maths > Set Theory > 4.0 Subsets
Set Theory
1.0 Set
2.0 Representation of set
3.0 Types of set
4.0 Subsets
5.0 Theorems based on Subsets
6.0 Power set
7.0 Subsets of the Set $R$
8.0 Intervals as subsets of $R$
9.0 Universal set
10.0 Venn Diagrams and operation on sets
11.0 Laws of algebra of sets
12.0 Number of elements in sets
4.1 Proper subset
A set $A$ is called a proper subset of $B$ if and only if each element of $A$ is an element of $B$ and there is at least one element of $B$ which is not in $A$.
i.e.
$$\begin{equation} \begin{aligned} \forall \;x \in A\;\;if\;x \in B\;and\;n(A) < n(B) \\ \Rightarrow A \subset B \\\end{aligned} \end{equation} $$
Illustration 13. State whether the following sets are the subsets of given sets ($M$).
a. $A = \{ a,b,c,d\} $ $M = \{ a,b,c,d,e,f,g\} $
b. $A = \{ x:x\;is\;a\;letter\,in\;the\,word\;CADRE\} $ $M = \{ x:x\;is\;not\;a\;letter\,in\;the\,word\;EDUCATION\} $
c. $A = \{ x:x = {n^2},x \in N\} $ $M = \{ x:x = {n^3},x \in N\} $
d.$A = \{ x:x = 4,x \in N\} $ $M = \{ x:x < {n^2},n < 10,n \in N\} $
e. $A = \{ x:10 < x < 35,x \in N\} $ $M = \{ x:10 < x < 35,x \in Z\} $
f. $A = \{ x:x\; < 10,x \in R\} $ $M = \{ x:x\; < 10,x \in N\} $
Solution:
| SET | CLASSIFICATION |
| a. $A = \{ a,b,c,d\} $ $M = \{ a,b,c,d,e,f,g\} $ | The elements of $A$ are, $a,b,c,d$ The elements of $M$ are, $a,b,c,d,e,f,g$ $\forall \;x \in A,\;x \in M$ Hence, $A \subset M$ |
| b. $B = \{ x:x\;is\;a\;letter\,in\;the\,word\;CADRE\} $ $M = \{ x:x\;is\;not\;a\;letter\,in\;the\,word\;EDUCATION\} $ | Elements of $B$ are, $A,C,D,E,R$ Elements of $M$ are, $A,C,D,E,I,N,O,T,U$ Clearly, $R \in B\;$ but, $R \notin M$ Hence, $B \not\subset M$ |
| c. $C = \{ x:x = {n^2},n \in N\} $ $M = \{ x:x = {n^3},n \in N\} $ | Elements of $C$ are, $1,4,9,16,......$ Elements of $M$ are, $1,8,27,64,......$ $\forall \;x \in C,\;x \notin M$ Hence, $C \not\subset M$ |
| d.$D = \{ x:x < 4,x \in N\} $ $M = \{ x:x < 7,x \in W\} $ | The elements of $D$ are, $1,2,3$ The elements of $M$ are, $0,1,2,3,4,5,6$ $\forall \;x \in D,\;x \in M$ Hence, $D \subset M$ |
| e. $E = \{ x:10 < x < 15,x \in N\} $ $M = \{ x:10 < x < 15,x \in Z\} $ | The elements of $E$ are, $11,12,13,14$ The elements of $M$ are, $11,12,13,14$ $\forall \;x \in E,\;x \in M$ and $\forall \;x \in M,\;x \in E$ Hence, $E \subseteq M$ |
| f. $F = \{ x:x\; < 10,x \in R\} $ $M = \{ x:x\; < 10,x \in N\} $ | Clearly, $F \not\subset M$, because, the set $F$ consists of real number, which include integers and decimals. Set $M$ consists of positive integers only. Thus, $\forall \;x \in F,\;x \notin M$. Hence, $F \not\subset M$ |
Illustration 14. Consider the following sets. Insert $ \subset ,\; \subseteq ,\; \not\subset $ between each of the pair of sets.
$A = \{ 2,4,6,8\} $ , $B = \{ 1,2,3,4,5\} $, $C = \{ 1,2,3,4,5,6,7,8\} $, $D = \{ 2,4\} $
| i. $A\_\_B$ | ii. $B\_\_A$ | iii. $A\_\_C$ | iv. $C\_\_A$ | v. $D\_\_A$ | vi. $A\_\_D$ |
| vii. $B\_\_C$ | viii. $C\_\_B$ | ix. $B\_\_D$ | x. $D\_\_B$ | xi. $C\_\_D$ | xii. $D\_\_C$ |
Solution:
i. $A\_\_B$ : Here, all elements of $A$ do not belong to $B$. Hence, $$A \not\subset B$$
ii. $B\_\_A$ : Here, all elements of $B$ do not belong to $A$. Hence, $$B \not\subset A$$
iii. $A\_\_C$ : Here, all elements of $A$ belong to $C$. Hence, $$A \subset C$$
iv. $C\_\_A$ : Here, all elements of $C$ do not belong to $A$. Hence, $$C \not\subset A$$
v. $D\_\_A$ : Here, all elements of $D$ belong to $A$. Hence, $$D \subset A$$
vi. $A\_\_D$ : Here, all elements of $A$ do not belong to $D$. Hence, $$A \not\subset D$$
vii. $B\_\_C$ : Here, all elements of $B$ belong to $C$. Hence, $$B \subset C$$
viii. $C\_\_B$ : Here, all elements of $C$ do not belong to $B$. Hence, $$C \not\subset B$$
ix. $B\_\_D$ : Here, all elements of $B$ do not belong to $D$. Hence, $$B \not\subset D$$
x. $D\_\_B$ : Here, all elements of $D$ belong to $B$. Hence, $$D \subset B$$
xi. $C\_\_D$ : Here, all elements of $C$ do not belong to $D$. Hence, $$C \not\subset D$$
xii. $D\_\_C$ : Here, all elements of $D$ belong to $C$. Hence, $$D \subset C$$
