Types of Sets - Commerce PDF Download

Types of Sets:
1. Finite Set- A set containing finite number of elements or no element.
2. Cardinal Number of a Finite Set- The number of elements in a given finite set is called cardinal number of finite set, denoted by n (A).
3. Infinite Set- A set containing infinite number of elements.
4. Empty/Null/Void Set- A set containing no element, it is denoted by (φ) or { }.
5. Singleton Set- A set containing a single element.
6. Equal Sets- Two sets A and B are said to be equal, if every element of A is a member of B and every element of B is a member of A and we write A = B.
7. Equivalent Sets- Two sets are said to be equivalent, if they have same number of elements. If n(A) = n(B), then A and B are equivalent sets. But converse is not true.
8. Subset and Superset- Let A and B be two sets. If every element of A is an element of B, then A is called subset of B and B is called superset of A. Written as A ⊆ B or B ⊇ A.
9. Proper Subset- If A is a subset of B and A ≠ B, then A is called proper subset of B and we write A ⊂ B.
10.Universal Set (U)- A set consisting of all possible elements which occurs under consideration is called a universal set.
11.Comparable Sets- Two sets A and Bare comparable, if A ⊆ B or B ⊆ A.
12.Non-Comparable Sets- For two sets A and B, if neither A ⊆ B nor B ⊆ A, then A and B are called non-comparable sets.
13.Power Set (P)- The set formed by all the subsets of a given set A, is called power set of A, denoted by P(A).
14.Disjoint Sets- Two sets A and B are called disjoint, if, A ∩ B = (φ).

Subset of a Set:
A set A is said to be a subset of the set B if each element of the set A is also the element of the set B. The symbol used is ‘⊆’ i.e.A ⊆ B ↔ ≤ x ∈A⇒ x ∈ B).

Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A ⊂ B. e.g If A = {a, b, c, d} and B = {b, c, d}. Then B⊂A or equivalently A⊃B ≤ i.e A is a super set of B). Total number of subsets of a finite set containing n elements is 2ⁿ.

Equality of Two Sets:
Sets A and B are said to be equal if A⊆B and B⊆A; we write A = B.

Some More Results Regarding the Order of Finite Sets:
Let A, B and C be finite sets and U be the finite universal set, then
i) n ≤ (A ∪ B) = n ≤ (A) + n ≤ B) – n ≤ (A ∩ B)
ii) If A and B are disjoint, then n ≤ (A ∪ B) = n ≤ (A) + n ≤ B)
iii) n ≤ A –B) = n ≤ (A) – n ≤ (A ∩ B) i.e. n ≤ (A) = n ≤ A – B) + n ≤ (A ∩ B)
iv) n ≤ A ∪ B ∪ C) = n ≤ (A) + n ≤ B) + n ≤ C) – n ≤ (A ∩ B) – n ≤ B ∩ C) – n ≤ A ∩ C) + n ≤ A ∩ B ∩ C)
v) n ≤ set of elements which are in exactly two of the sets A, B, C) = n ≤ A ∩B)+n ≤ B ∩ C) + n ≤ C ∩ (A) –3n≤ A ∩ B ∩ C)
vi) n≤ set of elements which are in atleast two of the sets (A, B, C) = n ≤ (A ∩ B) + n ≤ (A ∩ C) + n ≤ (B ∩ C) –2n≤ (A ∩ B ∩ C)
vii) n ≤ set of elements which are in exactly one of the sets (A, B, C)   = n ≤ (A) + n ≤ (B) + n ≤ (C) – 2n ≤ (A ∩ B) – 2n ≤ (B ∩ C) – 2n ≤ (A ∩ C) + 3n ≤( A ∩ B ∩ C)

Disjoint Sets:
If two sets A and B have no common elements i.e. if no element of A is in B and no element of B is in A, then A and B are said to be Disjoint Sets. Hence for Disjoint Sets A and B n ≤ (A ∩ B) = 0.

Illustration -:  If A and B be two sets containing 3 and 6 element respectively, what can be the minimum number of elements in A ∪ B? Find also, the maximum number of elements in A ∪ B.
Solution: We have, n ≤ (A ∪ B) = n≤ (A) + n≤ (B)  – n≤ (A ∩ B)

This shows that n ≤ (A ∪ B) is minimum or maximum according as

n ≤ (A ∩ B) is maximum or minimum respectively.
Case 1: When n ≤ (A ∩ B) is minimum, i.e. n ≤ (A ∩ B) = 0. This is possible only when A ∩ B = ϕ. In this case,

n≤ (A ∪ B) = n ≤ ((A)    + n ≤( B)  – 0 = n≤(A)    + n ≤ (B) = 3 +6 = 9

n ≤ (A ∪ B)_max = 9
Case 2: When n ≤ (A ∩ B) is maximum

This is possible only when A ⊆ B.

In this case n ≤ (A ∩ B) = 3

n ≤ (A∪B) = n≤ (A) + n≤ (B) – n ≤ (A ∩B) = ≤ (3+6-3)=6

n ≤ (A ∪ B)_min = 6.

Illustration -: In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How many can speak Hindi only? How many can speak Bengali? How many can speak both Hindi and Bengali?
Solution: Total number of people = 1000
n ≤ (H) = 750

n ≤ (B) = 400

n ≤ (H ∪ B) = n ≤ (H) + n ≤ (B) – n ≤ (H ∩ B)

n ≤ (H ∩ B) = 750 + 400 – 1000

= 150 speaking Hindi and Bengali both.

People speaking only Hindi = n ≤ (H) – n ≤ (H ∩ B) = 750 – 150 = 600

People speaking only Bengali = n ≤ (B) – n ≤ (H ∩ B) = 400 – 150 = 250.

Illustration-: A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, find the value of x.
Solution:  Let A denote the set of Americans who like cheese and let B denote those who like apples. Let the population of America be 100. Then,
n≤ (A) = 63, n ≤ (B)    = 76
Now, n ≤ (A ∪ B) = n≤ (A) + n≤ (B)    – n≤ (A ∩ B)
⇒ n≤ (A∪B) = 63+76-n ≤ (A ∩ B)
⇒ n ≤ (A ∩ B) = 139 – n ≤ (A ∪ B)
But n≤ (A∪B) ≤ 100 ⇒   n ≤ (A ∩ B ) = 39   …≤ i)
Now,    A  ∩ B ⊆ A and A ∩ B ⊆ B
⇒n≤ A ∩ B ) ≤ n ≤ (A)           and n ≤ (A ∩ B) ≤ n ≤ B)
⇒n ≤ (A ∩ B) ≤ 63  …≤ ii)
From  i) and  ii), we have 39 ≤ n ≤ (A ∩B ) ≤ 63 ⇒ 39 ≤ x ≤ 63.

Universal Set:
A non-empty set of which all the sets under consideration are subsets is called the universal set. In any application of set theory, all the sets under consideration will likely to be subsets of a fixed set called Universal Set. As name implies it is the set with collection of all the elements and usually denoted by ‘U’.
(e.g. ≤ 1) set of real numbers R is a universal set for the operations related to real numbers.

Set:
A set is a well-defined collection of distinct objects. Well-defined collection means that there exists a rule with the help of which it is possible to tell whether a given object belongs or does not belong to given collection. Generally sets are denoted by capital letters A, B, C, X, Y, Z etc.

Representation of a Set:
Usually, sets are represented in the following ways:
Roaster Form or Tabular Form:
In this form, we list all the member of the set within braces ≤  curly brackets and separate these by commas. For example, the set of all even numbers less than 10 and greater than 0 in the roster form is written as: A = {2,4, 6,8}
Set Builder Form or Rule Form:
In this form, we write a variable ≤ say x) representing any member of the set followed by a property satisfied by each member of the set.A = {x| x £ 5, x ∈ N} the symbol ‘|’ stands for the words” such that”.

Null/ Void/ Empty Set:
A set which has no element is called the null set or empty set and is denoted by ϕ  (phi). The number of elements of a set A is denoted as n  (A) and n ≤ ϕ) = 0 as it contains no element. For example the set of all real numbers whose square is –1.

Singleton Set:
A set containing only one element is called Singleton Set.

Finite and Infinite Set:
A set, which has finite numbers of elements, is called a finite set. Otherwise it is called an infinite set. For example, the set of all days in a week is a finite set whereas; the set of all integers is an infinite set.

Union of Sets:
Union of two or more sets is the set of all elements that belong to any of these sets. The symbol used for union of sets is ‘∪’ i.e. A∪B = Union of set A and set B = {x: x ∈ A or x∈B ≤ or both)}
Example: A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A∪B∪C = {1, 2, 3, 4, 5, 6, 8}

Intersection of Sets:

Similarly B – A = {x: x ∈ B and x ∉ A}
In general A-B ≠ B-A
Example: If A = {a, b, c, d} and B = {b, c, e, f} then A-B = {a, d} and B-A = {e, f}.

Symmetric Difference of Two Sets:
For two sets A and B, symmetric difference of A and B is given by  (A – B) ∪  (B – A) and is denoted by A Δ B.