Subsets
Let A and B be two sets. If every element of set A is also an element of set B, then A is called a subset of B.
We write A⊂ B, which is read as 'A is a subset of B'.

'A is a subset of B if x is an element of A implies that x is also an element of B' and this statement is denoted as A⊂B if x∈A⇒x∈B

Consider the sets A and B. B denotes the set of all students in your school and A denotes the set of all students in your class. Observe that every element of A is also an element of B. ∴ A⊂B

If there exist at least one element in A which is not a member of B, then A is not a subset of B and we write A⊄B.

Let A={1,2,3}, B={2,3,4,5,6}, C={1,2,3,4,5,6,7}
Here, A is subset of C. ∴ A⊂C
B is subset of C. ∴ B⊂C
A is not subset of B because, 1∈A and 1∉B. ∴ A⊄B

Superset
If A⊂B i.e. 'A is contained in B', we may also say that 'B contains A' or 'B is a superset of A'.
We write it as B⊃A , read as B is a superset of A.

Properties of Subsets
Every set is a subset of itself
Let A be any set, then each element of set A is clearly in set A. Hence, A⊂A.

The empty set is a subset of every set
Let A be any set and ϕ be empty set. ϕ contains no element. So every element of set ϕ is in set A. Hence, ϕ⊂A.

The total number of subsets of a finite set containing m elements is $\displaystyle \small 2^{m}$
Set Subsets No. of elements(m) No. of subsets
ϕ ϕ 0 $\displaystyle \small 2^{0}$ =1
{a} ϕ, {a} 1 $\displaystyle \small 2^{1}$=2
{a,b} ϕ, {a}, {b}, {a,b} 2 $\displaystyle \small 2^{2}$=4
{a,b,c} ϕ, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} 3 $\displaystyle \small 2^{3}$=8

Proper Subset
Let A be any set and B be a non-empty set, then A is called a proper subset of B if every member of A is also a member of B and there exists at least one element in B which is not a member of A.
If A is a proper subset of B, we write it as A⊂B, A≠B.
No. of proper subsets of any set with m number of elements=$\displaystyle \small 2^{m}$-1.

If A={b,c,d}, then proper subsets of A are ϕ, {b},{c},{d},{b,c},{c,d},{b,d}
No. of proper subsets =$\displaystyle \small 2^{3}$-1=7

{b,c,d} is not a proper subset because elements of this set are members of set A. 
A subset which is not a proper subset is called an improper subset.

A⊂B and B⊂A ⇔ A=B
If two sets A and B are equal i.e. A=B, then A⊂ B and B⊂A.
Conversely, if A⊂B and B⊂A, then A=B.
Thus A=B if and only if (denoted as⇔) for every a∈A⇒a∈B and for every b∈B⇒b∈A.

If A⊂B and B⊂C, then A⊂C
If x∈A, then x∈B since A⊂B
If x∈B, then x∈C since B⊂C
x∈A and x∈C Hence A⊂C

Example, Let A={x: x is a letter of the word 'BOWL'}, B={x: x is a letter of the word 'ELBOW'}, C={x: x is a letter of the word 'BELOW'}
Roaster forms are,
A={B,O,W,L}
B={E,L,B,O,W}
C={B,E,L,O,W}
(i) From roaster form we can say A⊂B, A⊂C, B⊂C, C⊂B, B⊄A, C⊄A, B=C

(ii)By properties of subset we can say, B⊂C and C⊂B, hence B=C
or B=C⇔B⊂C and C⊂B

(iii) A⊂B and B⊂C. Hence A⊂C
or A⊂C and C⊂B. Hence A⊂B

(iv)  A is a proper subset of B
A is a proper subset of C

(v) B is not proper subset of C
or C is improper subset of B

Power Set
The collection of all subsets of set A is called the power set of A.
It is denoted by P(A).
In P(A), every element is a set. ∴ P(A)={B: B⊂A}
If a finite set has m elements, then number of elements in power set=$\displaystyle \small 2^{m}$

Let A={1,2,3}, then P(A)={ϕ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
n(A)=3 and n[P(A)]=8

Universal Set
If there are some sets under consideration, then there happens to be a set which is a superset of each given set. Such a set is known as the universal set and is denoted by U or ξ.
It is kind of a parent set. Every set under discussion is a subset of universal set.

Let A={a,e,i,o,u}, B={b,c,d}, C={a,b,c,d,e,f} then,
U={x: x is a letter in English alphabet} or U={a,b,c,d,...,x,y,z} is the universal set.


Subsets of set of real numbers
N: set of all natural numbers. N={1,2,3,4,...}. N⊂W, N⊂Z, N⊂R, N⊄T
W: set of whole numbers. W={0,1,2,3,...}. W⊂Z, W⊂R
Z: set of all integers. Z={...,-3,-2,-1,0,1,2,3,4...}. Z⊂R
Q: set of all rational numbers. Q={x: x=p/q, p,q∈Z, q≠0}. Q⊂R
T: set of irrational numbers. St of real numbers that are not rational T={x: x∈R, x∉Q}. T⊂R
R: set of real numbers. R={x: x is a real number}. R⊂R
$\displaystyle \small Z^{+}$: set of positive integers. $\displaystyle \small Z^{+}$⊂Z⊂R
$\displaystyle \small Q^{+}$: set of positive rational numbers. $\displaystyle \small Q^{+}$⊂Q⊂R
$\displaystyle \small R^{+}$: set of positive real numbers. $\displaystyle \small R^{+}$⊂R

Intervals as subsets of R
Consider, A={x: 5≤ x≤9, x∈N}
This set can be easily represented as A={5,6,7,8,9}
Suppose, A={x: 5≤ x≤9, x∈R}
This set has infinite elements. So intervals are used to represent subsets of real numbers.

Let a,b belongs to R and a<b.
Then, the set of all real numbers lying between a and b, excluding the numbers a and b is said to form an open interval and is denoted by (a,b).
(a,b)={x: x∈R, a<x<b}

The set of all real numbers lying between a and b, including the numbers a and b is said to form a closed interval and is denoted by [a,b]
[a,b]={x: x∈R, a≤x≤b}

Some intervals are closed at one end and open at the other,
[a,b)={x: x∈R, a≤x<b} is an interval which includes a but excludes b.
(a,b]={x: x∈R, a<x≤b} is an interval which excludes a but includes b.

Examples (i) (2,10) is a subset of (-1,11) (ii) [5,8) is a subset of [5,8] (iii) (a,b] is a subset of [a,b] (iv) [0,∞ ) is the set of non negative real numbers (v) (-∞,∞) is the set of real numbers

Note
A={1,2,3,4,5,6} A={1,2,{3,4},5,6}
1∈A, 2∈A, 3∈A, 4∈A, 5∈A, 6∈A 1∈A, 2∈A, {3,4}∈A, 5∈A, 6∈A
n(A)=6 n(A)=5
{1}⊂A, {2}⊂A, {3}⊂A, {4}⊂A, {5}⊂A, {6}⊂A {1}⊂A, {2}⊂A, {{3,4}}⊂A, {5}⊂A, {6}⊂A
{1,2,3}⊂A,  {3,4,5,2}⊂A 3∉A, 4∉A, {3}⊄A, {4}⊄A, {3,4}⊄A, {1,2,3}⊄A, {1,2,{3,4}}⊂A
*set could be an element of a set. Here {3,4} is an element of set A.