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