A set is a well-defined collection of objects, whose elements are fixed and cannot vary. It means set doesn't change from person to person. Like for example, the set of natural numbers up to 7 will remain the same as {1,2,3,4,5,6,7}.
Types of Sets
Empty Sets
The set with no elements or null elements is called an empty set. This is also
called a Null set or Void set. It is denoted by {}.
For example: Let, Set X = {x:x is the number of students studying in Class 6th
and Class 7th}
Since we know a student cannot learn simultaneously on two classes, therefore
set X is an empty set.
Singleton Set
The set which has only one element is called a singleton set.
For example, Set X = { 2 } is a singleton set.
Finite and Infinite Sets
Finite sets are the one which has a finite number of elements, and infinite
sets are those whose number of elements cannot be estimated, but it has some
figure or number, which is very large to express in a set.
For example, Set X = {1, 2, 3, 4, 5} is a finite set, as it has a finite number
of elements in it.
Set Y = {Number of Animals in India} is an infinite set, as there is an
approximate number of Animals in India, but the actual value cannot be
expressed, as the numbers could be very large.
Equal Sets
Two sets X and Y are said to be equal if every element of set X is also the
elements of set Y and if every element of set Y is also the elements of set X.
It means set X and set Y have the same elements, and we can denote it as;
X = Y
For example, Let X = { 1, 2, 3, 4} and Y = {4, 3, 2, 1}, then X = Y
And if X = {set of even numbers} and Y = { set of natural numbers} the X ≠ Y,
because natural numbers consist of all the positive integers starting from 1,
2, 3, 4, 5 to infinity, but even numbers starts with 2, 4, 6, 8, and so on.
Subsets
A set X is said to be a subset of set Y if the elements of set X belongs to set
Y, or you can say each element of set X is present in set Y. It is denoted with
the symbol as X ⊂ Y.
We can also write the subset notation as;
X ⊂ Y if a ∊ X
a ∊ Y
Thus, from the above equation, “X is a subset of Y if a is an element of X
implies that a is also an element of Y”.
Each set is a subset of its own set, and a null set or empty set is a
subset of all sets.
Power Sets
The power set is nothing but the set of all subsets. Let us explain how.
We know the empty set is a subset of all sets and every set is a subset of
itself. Taking an example of set X = {2, 3}. From the above given statements we
can write,
{} is a subset of {2, 3}
{2} is a subset of {2, 3}
{3} is a subset of {2, 3}
{2, 3} is also a subset of {2, 3}
Therefore, power set of X = {2, 3},
P(X) = {{},{2},{3},{2,3}}
Universal Sets
A universal set is a set which contains all the elements of other sets.
Generally, it is represented as ‘U’.
For example; set X = {1, 2, 3}, set Y = {3, 4, 5, 6} and Z = {5, 6, 7, 8, 9}
Then, we can write universal set as, U = {1, 2, 3, 4, 5, 6, 7, 8, 9,}
Note: From the definition of the universal set, we can say, all the
sets are subsets of the universal set. Therefore,
X ⊂ U
Y ⊂ U
And Z ⊂ U
Union of sets
A union of two sets has all their elements. It is denoted by ⋃.
For example, set X = {2, 3, 7} and set Y = { 4, 5, 8}
Then union of set X and set Y will be;
X ⋃ Y = {2, 3, 7, 4, 5, 8}
Properties of Union of Sets:
X ⋃ Y = Y ⋃ X ;
Commutative law
(X ⋃ Y) ⋃ Z = X ⋃ (Y ⋃ Z)
X ⋃ {} = X ; {} is the identity of ⋃
X ⋃ X = X
U ⋃ X = U
Intersection of Sets
Set of all elements, which are common to all the given sets, gives
intersection of sets. It is denoted by the symbol ⋂.
For example, set X = {2, 3, 7} and set Y = {2, 4, 9}
So, X ⋂ Y = {2}
Difference of Sets
The difference of set X and set Y is such that, it has only those elements which
are in the set X and not in the set Y.
i.e. X – Y = {a: a ∊ X and a ∉ Y}
In the same manner, Y – X = {a: a ∊ Y and a ∉ X}
For example, if set X = {a, b, c, d} and Y = {b, c, e, f} then,
X – Y = {a, d} and Y – X = {e, f}
Disjoint Sets
If two sets X and Y have no common elements, and their intersection
results in zero(0), then set X and Y are called disjoint sets.
It can be represented as; X ∩ Y = 0
