Notes

Definition of a set:
A set is any well-defined collection of “objects”. The objects in the collection are called elements of the set.
Examples:
A set of all positive integers
A set of all the planets in the solar system
A set of all the states in India
A set of all the lowercase letters of the alphabet
The collection of pencils in your backpack is a set i.e. Each pencil in your backpack is an element of the set.

Notation:
Usually sets are denoted with upper-case letters, elements with lower-case letters.
Example: A = {1,2,3,4,5,6}

Representation of a Set
Sets can be represented in two ways −
Roster or Tabular Form
Set Builder Notation

Roster or Tabular Form:
The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.
Example 1 − Set of vowels in English alphabet, A = {a, e, i, o, u}

Set Builder Notation:
The set is defined by specifying a property that elements of the set have in common. Set builder notation has the general form {variable | descriptive statement}. The vertical bar (in set builder notation)(:) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.
The set is described as A={x : p(x)}
Example 1 − The set {a, e, i, o, u} is written as −
A = {x : x is a vowel in English alphabet}

Some Special Sets:
Null Set or Empty Set: This is a set with no elements, often symbolized by Ø
Universal Set: This is the set of all elements currently under consideration, and is often symbolized by ‘U’.

Cardinality of a Set:
Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.
Example − |{1, 4, 3, 5}| = 4 and n({1, 2, 3, 4, 5,…}) = ∞

If there are two sets X and Y,
=|X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ from X to Y.
=|X| ≤ |Y| denotes that set X’s cardinality is less than or equal to set Y’s cardinality. It occurs when number of elements in X is less than or equal to that of Y. Here, there exists an injective function ‘f’ from X to Y.
=|X| < |Y| denotes that set X’s cardinality is less than set Y’s cardinality. It occurs when number of elements in X is less than that of Y. Here, the function ‘f’ from X to Y is injective function but not bijective.
=If |X| ≤ |Y| and |X| ≥ |Y| then |X| = |Y|. The sets X and Y are commonly referred as equivalent sets.

There are three major operations performed on sets that are discussed below:
Union of sets (∪)
Intersection of sets (∩)
Difference of sets ( – )

If two sets A and B are given, then the union of A and B is equal to the set that contains all the elements, present in set A and set B. This operation can be represented as;
A ∪ B = {x: x ∈ A or x ∈ B}or
Where x is the elements present in both the sets A and B.
Example: If set A = {1, 2, 3, 4} and B = {6, 7}
Then, Union of sets, A ∪ B = {1, 2, 3, 4, 6, 7}

Difference of Sets
If there are two sets A and B, then the difference of two sets A and B is equal to the set which consists of elements present in A but not in B. It is represented by A-B.
Example: If A = {1,2,3,4,5,6,7} and B = {6,7} are two sets. Then, the difference of set A and set B is given by; A – B = {1,2,3,4,5}. We can also say, that the difference of set A and set B is equal to the intersection of set A with the complement of set B. Hence, A−B=A∩B’.

Intersection of Sets
If two sets A and B are given, then the intersection of A and B is the subset of universal set U, which consist of elements common to both A and B. It is denoted by the symbol ‘∩’. This operation is represented by:
A∩B = {x : x ∈ A and x ∈ B}and
Where x is the common element of both sets A and B.
The intersection of sets A and B, can also be interpreted as:
A∩B = n(A) + n(B) – n(A∪B)
where,
n(A) = cardinal number of set A,
n(B) = cardinal number of set B,
n(A∪B) = cardinal number of union of set A and B.
Example: Let A = {1, 2, 3} and B = {3, 4, 5}
Then, A∩B = {3}; because 3 is common to both the sets.


Sets can be classified into many types. Some of which are discussed as:
Finite Set: A set which contains a definite number of elements is called a finite set.
Example − S={x | x ∈ N and 90 > x > 50}

Infinite Set: A set which contains an infinite number of elements is called an infinite set.
Example − S = {x | x ∈ N and x > 10}

Subset: A set X is a subset of set Y (written as X ⊆ Y) if every element of X is an element of set Y. Here set Y is a subset of set X as all the elements of set Y is in set X. Hence, we can write Y ⊆ X. Here set Y is a subset (not a proper subset) of set X as all the elements of set Y is in set X. Hence, we can write Y ⊆ X.

Proper Subset: The term “proper subset” can be defined as “subset of but not equal to”. A Set X is a proper subset of set Y (written as X ⊂ Y) if every element of X is an element of set Y and |X| < |Y|. Here, set Y ⊂ X since all elements in Y are contained in X too and X has at least one element more than set Y.

Universal Set: It is a collection of all elements in a particular context or application. All the sets in that context or application are essentially subsets of this universal set. Universal sets are represented as U. In this case, set of all mammals is a subset of U, set of all fishes is a subset of U, set of all insects is a subset of U, and so on.

Empty Set or Null Set: An empty set contains no elements. It is denoted by ∅. As, the number of elements in an empty set is finite, empty set is a finite set. The cardinality of empty set or null set is zero.
Example − S= {x | x ∈ N and 7 < x < 8} = ∅

Singleton Set or Unit Set: Singleton set or unit set contains only one element. A singleton set is denoted by {s}.
Example − S = {x | x ∈ N, 7 < x < 9} = {8}

Equal Set: If two sets contain the same elements, they are said to be equal sets.

Equivalent Set: If the cardinalities of two sets are same, they are called equivalent sets. i.e. |A| = |B| = 3

Overlapping Set: Two sets that have at least one common element are called overlapping sets. In case of overlapping sets −
n(A∪B) = n(A) + n(B) − n(A∩B)
n(A∪B) = n(A − B) + n(B − A) + n(A∩B)
n(A) = n(A − B) + n(A∩B)
n(A) = n(A − B) + n(A∩B)
n(B) = n(B − A) + n(A∩B)

Disjoint Set: Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore, disjoint sets have the following properties −
n(A∩B) = ∅
n(A∪B) = n(A) + n(B)


What are Finite Sets?
Finite sets are sets having a finite or countable number of elements. It is also known as countable sets as the elements present in them can be counted. Finite sets can be easily represented in roster notation form. For example, the set of vowels in English alphabets, Set A = {a, e, i, o, u} is a finite set as the number of elements of the set are finite.

What are Infinite Sets?
Infinite sets can be understood as, sets that are not finite. The elements of infinite sets are endless, that is, infinite. If any set is endless from start or end or both sides having continuity then we can say that set is infinite. For example, the set of whole numbers, W = {0, 1, 2, 3, ……..} is an infinite set as the number of elements is infinite.


Difference Between Finite Sets and Infinite Sets

There are several similarities and differences between finite sets and infinite sets. Some of the common differences are summarized in the table below:

Finite Sets Infinite Sets

All finite sets are countable. Infinite sets can be countable or uncountable.

The union of two finite sets is finite. The union of two infinite sets is infinite.

A subset of a finite set is finite. A subset of an infinite set may be finite or infinite.

The power set of a finite set is finite. The power set of an infinite is infinite


Properties of Finite Sets
A proper subset of a finite set is finite.
The union of any number of finite sets is finite.
The intersection of two finite sets is finite.
The cartesian product of finite sets is finite.
The cardinality of a finite set is a finite number and is equal to the number of elements in the set.
The power set of a finite set is finite.

Properties of Infinite Sets
The union of any number of infinite sets is an infinite set.
The power set of an infinite set is infinite.
The superset of an infinite set is also infinite.
A subset of an infinite set may or may not be infinite.
Infinite sets can be countable or uncountable. For example, the set of real numbers is uncountable whereas the set of integers is countable.

Important Notes on Finite Sets and Infinite Sets:
An empty set is a finite set with cardinality equal to zero.
he cardinality of rational numbers is equal to the cardinality of natural numbers.
All finite sets are countable whereas infinite sets may or may not be countable


Cardinal numbers are counting numbers, so to find the cardinality of a set, the number of items in the set must be counted. This is a measurement of size or the number of elements within a specific set. The cardinality of a set is notated using the absolute value symbol. Here are some examples:
X = { }, therefore lXl = 0
W = {walrus}, therefore lWl = 1

How to Find Cardinality of a Set?
In order to determine the cardinality of a set, one must count the items in the set. A set can never contain a negative number of items. The cardinality will always be either zero, infinity, or a positive whole number. Here is one example :
T = {-64, -973, -35, -554}, therefore lTl = 4.

Power Set
The power set is a set which includes all the subsets including the empty set and the original set itself. It is usually denoted by P. Power set is a type of sets, whose cardinality depends on the number of subsets formed for a given set. If set A = {x, y, z} is a set, then all its subsets {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z} and {} are the elements of power set, such as:
Power set of A, P(A) = { {x}, {y}, {z}, {x, y}, {y, z}, {x, z}, {x, y, z}, {} }

Proper Power Set:
If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The symbol '⊂' is used to denote proper subset. Symbolically, we write A ⊂ B.

Cardinality of Power Set
Cardinality represents the total number of elements present in a set. In the case of a power set, the cardinality will be the list of the number of subsets of a set. The number of elements of a power set is written as |P (A)|, where A is any set. If A has ‘n’ elements then the formula to find the cardinality of power set (A) is given by:
|P(A)| = 2 ^ n = 2 raise to the power n
For example, set A = {1, 2, 3}
n = number of elements of A = 3
So, the number of subsets in a power set of A will be:
Subsets of A = {}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}
|P(A)| = 2^3 = 8
Hence, P(A) is {{}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}, {1,2,3,}}

Properties of Power Set
It is much larger than the original set.
The power set of a countable finite set is countable.
For a set of natural numbers, we can do one-to-one mapping of the resulted set, P(S), with the real numbers.
It exit in both the cases finite and infinite.

Power Set of Empty Set
An empty set has zero elements. Therefore, the power set of an empty set { }, can be mentioned as;
A set containing a null set.
It contains zero or null elements.
The empty set is the only subset.

Cartesian product of two sets:
Let A & B be any two sets. The set of all ordered pairs such that the first member of the ordered pair is an element of A & second member is an element of B is called the Cartesian product of A and B denoted as A × B.
That is, A × B ={(a, b) : (a є A) and ( b є B)}
A × (B × C) = {(a, (b, c)) : (a є A) and (b, c) є B × C}
A × B is not equals to B × A.
Example: Consider the sets A = {a, b} & B = {1, 2, 3}. Find A × B, B × A, and A × A .
Solution:
A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}
B × A = {(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)}
A × A = {(a, a), (a, b), (b, a), (b, b)}


Ordered pair:
An ordered pair of elements is written in the form (a, b) (or <a, b>) which is distinct from (b, a) unless a = b.
Note:
Equality of ordered pairs: (a, b) = (c, d) iff a = c & b = d
{1, 2} = {2, 1} = {1, 1, 2} but ordered pairs (1, 2), (2, 1) and (1, 1, 2) are not equal.
Ordered triple: Ordered triple can be represented as (a, b, c)
n-tuple: (x1, x2, x3, …, xn)