FOLLOW US:

Definition

The collection of well-defined distinct objects is known as a set. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not.

The word ‘distinct’ means that the objects of a set must be all different(Repeatition of elements are not allowed).

For example:

1 . The collection of children in class Xth whose weight exceeds 45 kg represents a set.

2. The collection of all the intelligent children in class X does not represent a set because the word intelligent is not clear. (in the case of intelligence there is no proper way to distinct, beacuse different people have different tendency)

---

Elements of Set:

The different objects that form a set are called the elements of a set. The elements of the set are written in any order (order has no meaning) and are not repeated. Elements are denoted by small letters.

---

---

Representations of a Set

Representation of Sets and its elements is done in the following two ways.

1 . Roster Form

In this form, all the elements are enclosed within braces {} and they are separated by commas (,). For example, a collection of all the numbers found on a dice N = {1, 2, 3, 4, 5, 6}. Properties of roster form: –

The order in which the elements are listed in the Roster form for any Set is immaterial. For example, V = {a, e, i, o, u} is same as V = {u, o, e, a, i}

The dots at the end of the last element of any Set represent its infinite form and indefinite nature. For example, group of odd natural numbers = {1, 3, 5, …}

In this form of representation, the elements are generally not repeated. For example, the group of letters forming the word POOL = {P, O, L}

More examples for Roster form of representation are:

A = {4, 8, 12, 16}

F = {2, 4, 8, 16, 32}

H = {1, 4, 9, 16, …, 100}

L = {3, 9, 27, 81}

Y = {1, 2, 3, 5, 8, …}

---

2 . Set Builder Form

In this form, all the elements possess a single common property which is NOT featured by any other element outside the Set. For example, a group of vowels in English alphabetical series.

The representation is done as follows. Let V be the collection of all English vowels, then – V = {x: x is a vowel in English alphabetical series.} Properties of Roster form: –

Colon (:) is a mandatory symbol for this type of representation.

After the colon sign, we write the common characteristic property possessed by ALL the elements belonging to that Set and enclose it within braces.

If the Set doesn’t follow a pattern, its Set builder form cannot be written.

More examples for Set builder form of representation for a Set: –

O = {y: y is a natural number greater than 5}

I = {f: f is a two – digit prime number less than 500}

X = {m: m is a positive integer < 100}

---

Types of Sets

Since, a Set is a well – defined collection of objects; depending on the objects and their characteristics, there are many types of Sets which are explained with proper examples, as follows: –

Finite and Infinite Sets

Any set which is empty or contains a definite and countable number of elements is called a finite set. Sets defined otherwise, for uncountable or indefinite numbers of elements are referred to as infinite sets. Examples:

A = {a, e, i, o, u} is a finite set because it represents the vowel letters in the English alphabetical series.

B = {x : x is a number appearing on a dice roll} is also a finite set because it contains – {1, 2, 3, 4, 5, 6} elements.

C = {p: p is a whole number} is an infinite set.

D = {k: k is a positive integer number} is also an infinite set.

---

Empty or Null or Void Set

Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – {} or f. Examples:

Let A = {x: 9< x < 10, x is a natural number} will be a null set because there is NO natural number between numbers 9 and 10. Therefore, A = {} or ∅

Let W = {d: d > 8, d is the number of days in a week} will also be a void set because there are only 7 days in a week.

---

Equal and Unequal Sets

Two sets X and Y are said to be equal if they have exactly the same elements (irrespective of the order of elements). Equal sets are represented as X = Y. Otherwise, the sets are referred to as unequal sets, which are represented as X ≠ Y. Examples:

If X = {a, e, i, o, u} and H = {u, o, i, a, e} then both of these sets are equal.

If C = {2, 12, 5, 7} and D = {1, 12, 5, 9} then both of these sets are unequal.

If A = {b, o, k} and B = {b, o, o, k, k} then also A = B because both contain same elements.

---

---

Singleton Set

These are those sets that have only a single element. Examples:

E = {x : x belongs to N and x³ = 64} is a singleton set with a single element {4}

W = {v: v is a even prime number} is also a singleton set with just one element {a}.

---

Equivalent Sets

Equivalent sets are those which have an equal number of elements irrespective of what the elements are. Examples:

A = {3,6,9,12,15} and B = {x : x is a vowel letter} are equivalent sets because both these sets have 5 elements each.

S = {1², 2², 3², 4², …} and T = {y : y² belongs to Natural number} are also equal sets.

---

Universal Set

A universal set contains ALL the elements of a problem under consideration. It is generally represented by the letter U. Example:

The set of even Numbers is a universal set

The study on Doctors, is conducted in whole U.P. then the list of all Doctors in U.P. is a universal set.

Power Set

The collection of ALL the subsets of a given set is called a power set of that set under consideration. Example:

A = {a, b} then Power set – P (A) = ∅, {a}, {b} and {a, b}. If n (A) = m (set A has m elements) then generally, n [P (A)] = 2^m

---

Subsets and Supersets

---

Operations on Sets

1 . Union

2 . Intersection

3 . Difference

4 . Complement

Let’s deal with them one by one.

Union of Sets

Let A = {2, 3, 6, 8} and B = {6, 8, 10, 12, 15}. Then, A U B is represented as the set containing all the elements that belong to both the sets individually. Mathematically,

A U B = {x : x ∈ A or x ∈ B}

So, A U B = {2, 3, 6, 8, 10, 12, 15},

here the common elements are not repeated.

Properties of (A U B)

Commutative law holds true as (A U B) = (B U A)

Associative law also holds true as (A U B) U {C} = {A} U (B U C)

Let A = {1, 2} B = {3, 4} and C = {5, 6}

A U B = {1, 2, 3, 4} and (A U B) U C = {1, 2, 3, 4, 5, 6}

B U C = {3, 4, 5, 6} and A U (B U C) = {1, 2, 3, 4, 5, 6}

Thus, the law holds true and is verified.

A U ∅ = A (Law of identity element)

Idempotent Law – A U A = A

Law of the Universal set (U): (A U U) = U

---

Intersection of Sets

An intersection is the collection of all the elements that are common to all the sets under consideration . Let A = {3, 4, 6, 8} and B = {3, 8, 10, 12} then A ∩ B or “A intersection B” is given by:

“A intersection B” or A ∩ B = {3, 8}

Mathematically, A ∩ B = {x : x ∈ A and x ∈ B}

Properties of the Intersection – A ∩ B

The intersection of the sets has the following properties:

Commutative law – A ∩ B = B∩A

Associative law – (A ∩ B)∩ C = A ∩ (B∩ C)

∅ ∩ A = ∅

U ∩ A = A

A∩A = A; Idempotent law.

Distributive law – A ∩ (BU C) = (A ∩ B) U (A ∩ C)

---

Difference of Sets

The difference of set A and B is represented as:

A – B = {x : x ∈ A and x ∈B}

Conversely, B – A = {x : x ∈ A and x ∈ B}

Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A – B = {1, 3, 5} and B – A = {8}. The sets (A – B), (B – A) and (A ∩ B) are mutually disjoint sets; it means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set.

---

Complement of Sets

If U represents the Universal set and any set A is the subset of A then the complement of set A (represented as A’) will contain ALL the elements which belong to the Universal set U but NOT to set A.

Mathematically, A’ = U – A

Alternatively, the complement of a set A, A’ is the difference between the universal set U and the set A.

Properties of Complement Sets

A U A’ = U

A ∩ A’ = ∅

De Morgan’s Law – (A U B)’ = A’∩ B’ OR (A ∩ B)’ = A’ U B’

Law of double complementation : (A’)’ = A

∅’ = U

U’ = ∅

---

De Morgan’s law

(A U B)’ = A’ ∩ B’ OR (A ∩ B)’ = A’ U B’