DMS - SETS

Set – Definition
A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any
element of a set is repeated, it does not make any changes in the set.

Some Example of Sets
* 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

Some Important Sets
N: the set of all natural numbers = {1, 2, 3, 4, .....}
Z: the set of all integers = {....., -3, -2, -1, 0, 1, 2, 3, .....}
Z+: the set of all positive integers
Q: the set of all rational numbers
R: the set of all real numbers
W: the set of all whole numbers

Cardinality of a Set
Cardinality of a set S, denoted by |S|, is the number of elements of the set. If a set has an infinite number of elements, its cardinality is ∞.
Example: |{1, 6, 3}| = 3, |{1, 2, 3,4,5,…}| = ∞

Finite Set
A set which contains a definite number of elements is called a finite set.
Example: S = {x | x ∈ N and 50 > x > 20}

Infinite Set
A set which contains infinite number of elements is called an infinite set.
Example: S = {x | x ∈ N and x > 20}

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.
Example 1: Let, X = { 1, 2, 3, 4, 5, 6 } and Y = { 1, 2,3 }. Here set X is a subset of
set Y as all the elements of set X is in set Y. Hence, we can write X ⊆Y.

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 |.
Example: Let, X = {1, 2,3,4,5, 6} and Y = {1, 2,3}. Here set X is a proper subset of set Y as at least one element is more in set Y. Hence, we can write X ⊂ 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.
Example: We may define U as the set of all animals on earth. 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: ∅ = {x | x ∈ N and 8 < x < 9}

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, 8 < x < 10}

Equal Set
If two sets contain the same elements they are said to be equal.
Example: If A = {2, 4, 7} and B = {7, 2, 4}, they are equal as every element of set A is an element of set B and every element of set B is an element of set A.

Equivalent Set
If the cardinalities of two sets are same, they are called equivalent sets.
Example: If A = {1, 2, 3, 7} and B = {15, 17, 18, 23}, they are equivalent as cardinality of A
is equal to the cardinality of B. i.e. |A|=|B|=4.

Disjoint Set
If two sets C and D are disjoint sets as they do not have even one element in common.
Therefore, n(A ∪ B) = n(A) + n(B)
Example: Let, A = {11, 12, 116} and B = {8, 19, 114}, there is no common element, hence
these sets are overlapping sets.

Cartesian Product / Cross Product
The Cartesian product of n number of sets A1, A2.....An, defined as A1 × A2 ×..... × An, are the ordered pair (x1,x2,....xn) where x1∈ A1 , x2∈ A2 , ...... xn ∈ An
Example: If we take two sets A= {a, b} and B= {1, 2},
The Cartesian product of A and B is written as: A×B= {(a, 1), (a, 2), (b, 1), (b, 2)}
The Cartesian product of B and A is written as: B×A= {(1, a), (1, b), (2, a), (2, b)}

Power Set
Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is 2n. Power set is denoted as P(S).
Example:
For a set S = {a, b, c, d} let us calculate the subsets:
* Subsets with 0 elements: {∅} (the empty set)
* Subsets with 1 element: {a}, {b}, {c}, {d}
* Subsets with 2 elements: {a,b}, {a,c}, {a,d}, {b,c}, {b,d},{c,d}
* Subsets with 3 elements: {a,b,c},{a,b,d},{a,c,d},{b,c,d}
* Subsets with 4 elements: {a,b,c,d}
Hence, P(S) = { {∅},{a}, {b}, {c}, {d},{a,b}, {a,c}, {a,d}, {b,c}, {b,d},{c,d},{a,b,c},{a,b,d},{a,c,d},{b,c,d},{a,b,c,d} }
| P(S) | = 24 =16
Note: The power set of an empty set is also an empty set.
| P ({∅}) | = 2pow0 = 1

Partitioning of a Set
Partition of a set, say S, is a collection of n disjoint subsets, say P1, P2,...… Pn, that satisfies the following three conditions:
* Pi does not contain the empty set.
[ Pi ≠ {∅} for all 0 < i ≤ n]
* The union of the subsets must equal the entire original set.
[P1 ∪ P2 ∪ .....∪ Pn = S]
* The intersection of any two distinct sets is empty.
[Pa ∩ Pb ={∅}, for a ≠ b where n ≥ a, b ≥ 0 ]
The number of partitions of the set is called a Bell number denoted as Bn.
Example
Let S = {a, b, c, d, e, f, g, h}
One probable partitioning is {a}, {b, c, d}, {e, f, g,h}
Another probable partitioning is {a,b}, { c, d}, {e, f, g,h}
In this way, we can find out Bn number of different partitions.


Venn Diagrams
Venn diagram, invented in1880 by John Venn, is a schematic diagram that shows all possible logical relations between different mathematical sets.

 

Venn Diagram Examples

Set Operations

Set Operations include Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product.

Set Union
The union of sets A and B (denoted by A ∪ B) is the set of elements which are in A, in B, or in both A and B. Hence, A∪B = {x | x ∈A OR x ∈B}.
Example: If A = {10, 11, 12, 13} and B = {13, 14, 15}, then A ∪ B = {10, 11, 12, 13, 14, 15}. (The common element occurs only once)

 Venn Diagram of AUB

Set Intersection

The intersection of sets A and B (denoted by A ∩ B) is the set of elements which are in
both A and B. Hence, A∩B = {x | x ∈A AND x ∈B}.
Example: If A = {11, 12, 13} and B = {13, 14, 15}, then A∩B = {13}.

Set Difference/ Relative Complement
The set difference of sets A and B (denoted by A–B) is the set of elements which are only in A but not in B. Hence, A−B = {x | x ∈A AND x ∉B}.
Example: If A = {10, 11, 12, 13} and B = {13, 14, 15}, then (A−B) = {10, 11, 12} and (B−A) = {14,15}. Here, we can see (A−B) ≠ (B−A)

Complement of a Set

The complement of a set A (denoted by A’) is the set of elements which are not in set A. Hence, A' = {x | x ∉A}.
More specifically, A'= (U–A) where U is a universal set which contains all objects.
Example: If A ={x | x belongs to set of odd integers} then A' ={y | y does not belong to set of odd integers}