A Set is a collection of distinct and defined elements. Sets are represented by using French braces {} with commas to separate the elements in a Set.

There are various types of sets as follows:
• Finite Set
• Infinite Set
• Empty or Null Set
• Equal Sets
• Disjoint Sets
• Intersecting Sets

### Examples of Sets

• The set of all points on a particular line.
• The set of all lines in a particular plane.
• A set can also contain elements which are themselves sets. For example, a set may contain 4, 5, {6, 7}.

## Set Operations

There are three basic Set Operations.
• Intersection of Sets
• Union of Sets
• Complement of Sets

Properties on Operations of Sets

A ⊂ B, B ⊂ C ⇒ A ⊂ C (Property of Transitivity)

A ⊂ B, B ⊂ A ⇒ A = B

A ∪ A = A

A ∩ A = ∅

A ∪ ∅ = A

A ∩ ∅ = ∅

A ∪ B = B ∪ A (Commutative law for addition)

A ∩ B = B ∩ A (Commutative law for multiplication)

(A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative law for addition)

(A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative law for multiplication)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law for addition)

A ∪ (B ∩ C) = (A ∩ B) ∪ (A ∩ C) (Distributive law for multiplication)

## Set Basic Concept

The concept of a set is of fundamental importance in Mathematics. A football team is a set of players. A class is a set of students. The school library houses a set of books on Mathematics, a set of books on Physics, and so on.

Thus, we can say that a set is a well defined collection of objects. When we say "well-defined" it means that we must be given a rule or rules with the help of which we should be readily able to say whether a particular object is a member of the set or or not.

Another example, vowels in the English alphabet form a set because any of the alphabet is either a vowel or a consonant. The collection of all honest people in a country is not a set, because the term "honest" is not well-defined.

If S is any set, every object in S is called an element of the set. For example, let S be the set of all natural numbers less than 100. Then, S = {1, 2, 3, 4, ......., 10, 11, ........97, 98, 99}

10 is an element of the set. Also, 36 is an element of the set. The fact that 10 is an element of the set expressed in symbols as 10 ∈ S which is read as "10 belongs to S" or "10 is an element of S".

150 is not an element of the set S. We represent it as 150 ∉ S which is read as "150 does not belong to S" or "150 is not an element of S".

The elements of a set are generally denoted by small letters a, b, c... x ,y, z. The sets are denoted by capital letters A, B, C ........, X, Y, Z.

In general,
• If an element x is in set A, then we say x belongs to A and we write x ∈ A.
• If an element x is not in set A, then we say x does not belongs to A and we write x ∉ A.

## Types of Sets

There are various types of sets:

• Finite Set
• Infinite Set
• Null Set
• Singleton Set
• Equivalent Sets
• Equal Sets

### Finite Set

A set is said to be finite if it contains only finite number of elements.

Let A = {1, 2, 3, 4, 5}
B = {a, b, c, d, e, f}
Here, A contains 5 elements and B contains 6 elements. So, A and B are finite sets

### Examples of Finite Set

• The set {2, 4, 6} Is a finite set, as it contains only 3 elements.
• If A is the set of all days in a week, then A is a finite set containing 7 elements.

### Infinite Set

A set is said to be infinite if it contains an infinite number of elements.

Let C = {number of men living in different parts of the world}
It is difficult to find the number of elements in C. But, it is a definite number, may be quite a big number. And so, C is an infinite set.

### Examples of Infinite Set

• The set of all even numbers is an infinite set.
• The set of points on a particular straight line is an infinite set.
• The sets N (Natural Numbers), Z (Integers), Q (Rational Numbers), R (Real Numbers) and C(Complex Numbers) are all infinite sets, where:
1. N = { 1, 2, 3, 4, .......................}
2. Z = {.............., -3, -2, -1, 0, 1, 2, 3, ..........}
3. Q = {$\frac{(p)}{(q)}$: p, q in Z, q ∉ 0}
4. R = { x : " x is either a rational number or an irrational number "}
5. C = { x + iy : x, y in R, i = $\sqrt{-1}$}

### Null Set

A set is said to be a null set if it does not contain any element. A null set is also called as an empty set or void set. A null set is denoted by Φ.
Therefore, Φ = { }
The set { 0 } is not a null set, because this set contains one element "0".

### Examples of Null Set

• Let A = { x : x ∈ N, 2 < x <3}. A does not contain any element , because there is no natural number between 2 and 3.
• Let B = { x : x ∈ Q, 2 < x <3}. B is not a null set, because rational numbers like $\frac{5}{2}$, $\frac{7}{3}$, $\frac{9}{4}$, $\frac{11}{5}$ , ....... are all elements of the set B.

### Singleton Set

A set is said to be a singleton set, if it contains only one element.

### Examples of the Singleton Set

• The set {7} , {-15} are singleton sets.
• {x : x + 4 = 0, x ∈ Z} is a singleton set, because this set contains only one integer namely, -4.

### Equivalent Sets

Two sets A and B are said to be equivalent sets if the elements of A can be paired with the elements of B, so that each element of A corresponds exactly to one element of B, and each element of B there corresponds exactly to one element of A.

### Examples of Equivalent Sets

• The sets {a, b, c} and {4, 7, 10} are equivalent.
• The sets {w, x, y, z} and {1, 2, 3, 4} are equivalent.

### Equal Sets

Two sets are said to be equal sets if every element of one set is in the other set and vice-versa. So, two sets are equal, if x in A ⇒ x in B and x ∈ B ⇒ x ∈ A. If sets A and B are not equal, then we write A ≠ B

### Examples of Equal Sets

• Let A = {x : x ∈ N, 2 x 6 } and B = { 2, 3, 4, 5, 6} , then A = B
• Let A = {x : x ∈ N, 10< x<11} and B = {10.5} , then A ≠ B since 10.5 ∉ A

## Intersection of Sets

The Intersection of two or more sets is a set containing only the common elements among all the sets under consideration. The intersection operation is denoted by the symbol ∩. The Intersection of the sets A and B is expressed as A ∩ B.

### Examples on Intersection of Sets

Given below are some examples that explain the intersection of sets.

Example 1:

If A = {Cat, Dog, Mouse, Lion, Tiger} and B = {Cat, Lion, Elephant, Tiger}, Find A ∩ B.

Solution:

A = {Cat, Dog, Mouse, Lion, Tiger}
B = {Cat, Lion, Elephant, Tiger}

A ∩ B = {Cat, Lion, Tiger}

Example 2:

If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 7, 9}, Find A ∩ B.

Solution:

A = {1, 2, 3, 4, 5, 6, 7, 8}
B = {1, 3, 5, 7, 9}

A ∩ B = {1, 3, 5, 7}

## Union of Sets

The Union of two or more sets is a set that contains all the elements of all the sets under consideration. The union of sets is denoted by the symbol ∪. The Union of sets C and D is expressed by C ∪ D.

### Examples on the Union of Sets

Given below are some of the examples that explain the union of sets.

Example 1:

If A = {Cat, Dog, Mouse, Lion, Tiger} and B = {Cat, Lion, Elephant, Tiger}, Find A ∪ B.

Solution:

A = {Cat, Dog, Mouse, Lion, Tiger}
B = {Cat, Lion, Elephant, Tiger}

A ∪ B = {Cat, Dog, Mouse, Lion, Tiger, Elephant}

Example 2:

If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 7, 9}, Find A ∪ B.

Solution:

A = {1, 2, 3, 4, 5, 6, 7, 8}
B = {1, 3, 5, 7, 9}

A ∩ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

## Complement of Sets

The Complement of a set is a set that contains only the elements which are not in the given set but are contained in the universal set. The Complement of a set is denoted by A'.

• The Complement of an Universal Set is a Null Set. That is, U' = ∅
• The Complement of a Null Set is an Universal Set. That is, ∅' = U

### Examples on the Complement of Sets

Given below are some examples that explain the complement of sets.

Example 1:

If U = {Red, Blue, Green, Yellow, Violet} and A = {Blue, Yellow}, Find complement of A.

Solution:

The Complement of A will have those elements which are not in A but are in U.

So, A' = {Red, Green, Violet}

Example 2:

If U = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 7}, Find A'.

Solution:

U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 3, 5, 7}

A' = {2, 4, 6, 8}

## Cardinal Number of the Set

Cardinality is a measure of a size of a set or simply the total number of elements in a set.

### Examples of Cardinal Number of a Set

• Cardinality of a set of vowels is 5 as it has 5 elements (a, e ,i , o, u).
• Cardinality of a Set of the number of months in a year is 12 as it has 12 elements (January, February, ......., December).

Notation: Cardinality of a set is denoted by |A| or #A. The Cardinality of a set of natural numbers is denoted by N0 and that of the Real numbers is C.

### Properties of Cardinal number or Cardinality of Sets

• Two Sets have the same Cardinal number if the function from A to B is Bijective. That is, injective (one-one) as well as surjective (onto). That is |A| = |B|.

Example: If A = {0, 1, 2, 3,........} and B = {0, 5, 10,.........}, then the function can be defined as, f(a) = b (where, a and b represents the elements of Sets A and B). So, the function is Bijective. Hence, |A| = |B|.

• Cardinal number of a set A is less than or equal to that of set B, if the function from A to B is injective but not surjective. That is, |A| ≤ |B|.

Example: If A = {1, 2, 3, 4, 5, ..........} and B = {0, 1, 2, 3, 4, ..........} and if a function is defined from A to B such that f(a) = a, then the element 0 of Set B does not have a pre-image in A. So, the function is Injective but not Surjective. Hence, |A| ≤ |B|.

Cardinality helps in determining the types of Sets.

• Any set is a finite set, if its cardinality is less than that of natural number. That is, |A| ≤ |N|
• Any set is a countably infinite set, if its cardinality is equal to that of the natural number. That is, |A| = |N|
• Any set is an uncountable set, if its cardinality is greater than that of the natural number. That is, |A| ≥ |N|

## Set Theory Examples

Given below are some solved examples in the set theory.

Example 1:

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {2, 4, 6, 8, 10}, B = (1, 3, 6, 7, 8} and C = {3, 7}, find A ∩ B, A ∪ C, B ∩ A′, B ∩ C′

Solution:

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
A = {2, 4, 6, 8, 10}
B = (1, 3, 6, 7, 8}
C = {3, 7}

A ∩ B = {6, 8}
A ∪ C = {2, 3, 4, 6, 7, 8, 10}
B ∩ A′ = {1, 3, 7}
B ∩ C′ = {1, 6, 8}

Example 2:

If U = {Pencil, Pen, Eraser, Notebook}, P = {Pencil, Notebook} and Q = {Pen, Eraser}, find P ∪ Q, P ∩ Q, P ∪ Q', P ∩ Q'

Solution:

U = {Pencil, Pen, Eraser, Notebook}
P = {Pencil, Notebook}
Q = {Pen, Eraser}

P ∪ Q = {Pencil, Notebook, Pen, Eraser} = U
P ∩ Q = ∅
P ∪ Q' = {Pen, Eraser, Pencil, Notebook}
P ∩ Q' = ∅

Example 3:

At a breakfast buffet, 93 people preferred coffee as a beverage, 47 people preferred juice, 25 preferred both coffee and juice. If each person prefers atleast one of the beverages, then how many people visited the buffet?

Solution:

Let A be the set of people who prefer coffee and B be the set of people who prefer juice.

n(A) = 93, n(B) = 47, n(A ∩ B) = 25

n(AUB) = n(A) + n(B) - n(A ∩ B)

Plugging-in all the values,

n(AUB) = 93 + 47 - 25
n(AUB) = 115

Hence, the number of people who visited the buffet is 115.

Example 4:

In a class of 50 students, 30 speak Spanish, 15 speak both Spanish and English. How many students speak English?

Solution:

Let A be the Set of students who speak Spanish and B be the Set of students who speak English.

n(AUB) = 50, n(A) = 30, n(A ∩ B) = 25
n(AUB) = n(A) + n(B) - n(A ∩ B)

Plugging-in all the values,
50 = 30 + n(B) - 15
n(B) = 35

Hence, the number of students who speak English is 35.

## Laws of Set Theory

Given below are some laws of set theory:

 1 ¬(¬A) = A Law of Double Complement 2 ¬(A ∪ B) = ¬A ∩ ¬B ¬(A ∩ B) = ¬A ∪ ¬B DeMorgan's Laws 3 A ∪ B = B ∪ A A ∩ B = B ∩ A Commutative Laws 4 A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C Associative Laws 5 A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Distributive Laws 6 A ∪ A = A A ∩ A = A Idempotent Laws 7 A ∪ ∅ = A A ∩ U = A Identity Laws 8 A ∪ ¬A = U A ∩ ¬A = ∅ Inverse Laws 9 A ∪ U = U A ∩ ∅ = ∅ Domination Laws 10 A ∪ (A ∩ B) = A A ∩ (A ∪ B) = A Absorption Laws 11 A - B = A ∩ ¬B Definition of Set Difference 12 A Δ B = (A ∪ B) - (A ∩ B) Definition of Symmetric Difference

## Subsets

A Set is called as a subset of another set if all it's elements are contained in another set. The symbol used for the subset is ⊂. The set A is a subset of the set B and is expressed as A ⊂ B. Each non empty set has at least two subsets - Empty Set (or Null Set) and the set itself. The number of subsets of a set is calculated by the formula 2n, where n is the number of elements of the Set.

Empty Set or Null Set is represented by either {} or Φ (read as phi)

### Examples on Subsets

Given below are some examples on subsets.

Example 1:

S = {a, b, c}
Here, the number of elements is 3.
So, the total number of Subsets is 23 = 8.
The Subsets are: Φ , {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}.

Example 2:
E = Φ (an empty set)
Here, the number of elements is 0.
So, the total number of Subsets is 20 = 1.
The only Subset is: Φ

## Proper Subset

A set A is said to be a proper subset of a set B, if A is a subset of B and A is not equal to B. If A is a proper subset of B, then we write A ⊂ B. If A is a proper subset of B, then B must have at least one element which is not in A.

### Examples of Proper Subset

Given below are some examples of proper subset.

Example 1:

Let B be the set of closed objects as shown below:

Let us consider the set A as follows:

Then, clearly every object in A is an object in B. Hence, A ⊂ B

Example 2:

The set of natural numbers, N ⊂ Z, the set of integers, because -2 ∈ Z and -2 ∉ N. If A = {1, 2, 3} , then proper subsets are Φ, {1}, {2}, {3}, {1,2}, {2,3}, {1,3}

If A ⊆ B, then every element of x in A is in B and there is a chance that A may be equal to B, that is, every element of B is in A. But, in case A ⊂ B, then every element of A is in B and there is no chance that A may be equal to B, that is, there will exist at least one element in B which is not in A.

### Properties of Proper Subset

• If a set has only one element, then it has two subsets.
• If B sub A and if A has one element more than B, then A has twice as many subsets as B.
• A set with two elements has 22 subsets, A set with three elements has 23 subsets, and so on. Hence a set with n elements has 2n subsets.

## Disjoint Sets

Two sets A and B are called disjoint sets if there is no element which is both in A and B. Disjoint sets can be also defined as those sets which do not overlap or are not duplicated. Let us consider two sets as follows:

Let A be the set of triangles as shown

Let B be the set of arrows as shown

Here, the set A and B have no elements in common. So, A ∩ B = Φ. Hence, such sets are called as disjoint sets.

### Examples of Disjoint Sets:

Given below are some examples of disjoint sets.

Example 1:

Let A ={ 4, 6, 10} and B = {7, 11, 15}
Since there is no number common between A and B, we get A ∩ B = Φ
So, A and B are disjoint.

Example 2:

The set of intervals { [1, 3], [2,5], (7,9)} is not disjoint, since [1, 3] is overlapping [2,5]. Disjoint sets are also said to be mutually exclusive or independent.

Disjoint sets can be represented by using venn-diagram as:

### Mutually Disjoint Sets

Set A1, A2, A3, ......, An are mutually disjoint if Ai ∩ Aj = Φ for i ≠ j. A partition of set A is a collection of subsets Ai of A such that Ai ≠ Φ , A1, A2, A3, .........., An are mutually disjoint and ∪ Ai = A

For example, let the universal set be U = {1, 2, 3, 4, 5}
and let its subsets be A = {1, 2, 3} and B = {4, 5}.
Then, A ∩ B = Φ and A ∪ B = U. So, A and B are partitions of U

### Line of Symmetry

 Venn Diagram Cartesian Product Set Difference Subset Set Operations
 Set of even Numbers Solution Sets Probability Theory Bivariate Data Set Cardinal Number of a Set Cartesian Product of Two Sets Categorical Data Sets Correlation Data Set Data Sets for Statistics Set of Counting Numbers Concentric Circle Theory Two Step Algebraic Equations
 Determinant Calculator 2x2 eigenvalue calculator 2x2 Difference of Two Squares Calculator