proper and improper subset example

subset formula


Title: Proper and Improper Subset Example

A proper subset is a set that contains some, but not all, of the elements of another set. For instance, if we have set A {1, 2, 3} and set B {1, 2, 3, 4}, then A is a proper subset of B. This is because all elements of A are also present in B, but B contains an additional element. Conversely, an improper subset is a set that contains all the elements of another set. In the example above, B is an improper subset of itself, as it contains all the elements of B.

Proper and improper subset example

proper subset:

All subset of a set is called proper subset without itself set. In more detail, If B is a subset of C and C contain at least one element which is not an element of B, then B is said to be a proper subset of C.

In such a case, we write B ⊂ C (B is a proper subset of C).

Proper subset formula:⇓

proper subset formula


In  mathematically proper subset is denoted by


Improper subset: 

Every  set is called itself improper subset


C= {4,7,9}  is called improper subset without other subset.

From this definition, it also follows that every set C is an improper subset of itself.

Proper subset formula:

 proper subset formula

if a set has element B= {2,4,6} where n=3

applying this formula

proper subset formula applying method

there are seven proper subsets

Example of proper and improper subset:

we take three set A, B, C

where set A ={3,4,5}    B= {5,3,4} and C= {3,4,5,6}

A ⊂ C(A is subset of C),  B ⊂ C(b is subset of C) ,  and B = A

notice that each number of set  A and B is an improper subset of the other

Because A=B

set A= {3,4,5}

proper subset = {}, {3}, {4}, {5}, {3,4}, {3,5}, {4,5}

improper subset = {3,4,5}

Important note:

when we do not want to distinguish between the proper and improper set

we may use the symbol ⊆  (subset) for the relationship. It is easily to say that

N ⊂ W ⊂ Z ⊂ Q ⊂ R

Subset means:

A set is a combination of element or object group in curly bracket such that {5,6,7,7,9}. If a set C is a combination of prime numbers and set D is {1,3,5} then we say that D is a subset of C and mathematically denoted by D ⊆ C.

Subset formula

subset formula

Number of subset formula:   all subset in the set

How many subsets in set  we calculate this formula. If we have number of element in the set is n=2

\large 2^{n}\Rightarrow 2^{2}=4

number of subset in the set is 4

Element of subset:

The element of subset can be natural numbers, integers, number, rational number, complex numbers, or real number, and complex numbers iota anything in the world can be performed in the set.

Subsets symbols:

In branch of mathematic set theory  subset symbol denoted by (read is a subset of).

If set A  is greater than set B (mathematically > notation greater than)

we write this B ⊆ A (B is subset of A)

 all subset of set:

All subset consisting on the set with null subset and itself subset, itself subset is itself set.

We understand with the help of example.


B= {2,3,4} fined all subset of a set

{}, {2}, {3}, {4}, {2,3}, {2,4}, {3,4},{2,3,4}

these subsets have

proper subset ={}, {2}, {3}, {4}, {2,3}, {2,4}, {3,4}

improper subset ={2,3,4}

Empty set is proper or improper:

Empty set is subset of itself and improper subset of itself, otherwise in group of subset is proper subset.