Discrete Ass
Discrete Ass
Discrete Ass
Sets can be classified into many types. Some of which are finite, infinite, subset, universal, proper, singleton set, etc.
Finite Set
Infinite Set
Subset
Example 1 − Let, X={1,2,3,4,5,6} X={1,2,3,4,5,6} and Y={1,2}Y={1,2}. Here set Y is a subset of set X as all the
elements of set Y is in set X. Hence, we can write Y⊆XY⊆X.
Example 2 − Let, X={1,2,3} X={1,2,3} and Y={1,2,3} Y={1,2,3}. Here set Y is a subset (Not a proper subset) of
set X as all the elements of set Y is in set X. Hence, we can write Y⊆XY⊆X.
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⊂YX⊂Y) if every element of X is an element of set Y and |X|<|Y||X|<|Y|.
Example − Let, X={1,2,3,4,5,6} X={1,2,3,4,5,6} and Y={1,2} Y={1,2}. Here set Y⊂XY⊂X since all elements
in YY are contained in XX too and XX has at least one element is more than set YY.
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 UU.
Example − We may define UU as the set of all animals on earth. In this case, set of all mammals is a subset
of UU, set of all fishes is a subset of UU, set of all insects is a subset of UU, and so on.
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.
Singleton set or unit set contains only one element. A singleton set is denoted by {s}{s}.
Example − S={x|x∈N, 7<x<9}S={x|x∈N, 7<x<9} = {8}{8}
Equal Set
If two sets contain the same elements they are said to be equal.
Example − If A={1,2,6}A={1,2,6} and B={6,1,2}B={6,1,2}, 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.
Overlapping Set
Two sets that have at least one common element are called overlapping sets.
n(A∪B)=n(A)+n(B)−n(A∩B)n(A∪B)=n(A)+n(B)−n(A∩B)
n(A∪B)=n(A−B)+n(B−A)+n(A∩B)n(A∪B)=n(A−B)+n(B−A)+n(A∩B)
n(A)=n(A−B)+n(A∩B)n(A)=n(A−B)+n(A∩B)
n(B)=n(B−A)+n(A∩B)n(B)=n(B−A)+n(A∩B)
Example − Let, A={1,2,6} A={1,2,6} and B={6,12,42} B={6,12,42}. There is a common element ‘6’, hence these
sets are overlapping sets.
Disjoint Set
Two sets A and B are called disjoint sets if they do not have even one element in common. Therefore,
disjoint sets have the following properties −
n(A∩B)=∅n(A∩B)=∅
n(A∪B)=n(A)+n(B)n(A∪B)=n(A)+n(B)
Example − Let, A={1,2,6}A={1,2,6} and B={7,9,14}B={7,9,14}, there is not a single common element, hence
these sets are overlapping sets.
Venn Diagrams
Venn diagram, invented in 1880 by John Venn, is a schematic diagram that shows all possible logical
relations between different mathematical sets.
Examples
Set Operations
Set Union
The union of sets A and B (denoted by A∪BA∪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} A∪B= {x|x∈A OR x∈B}.
Set Intersection
The intersection of sets A and B (denoted by A∩BA∩B) is the set of elements which are in both A and B.
Hence, A∩B={x|x∈A AND x∈B}A∩B={x|x∈A AND x∈B}.
The set difference of sets A and B (denoted by A–BA–B) is the set of elements which are only in A but not in B.
Hence, A−B={x|x∈A AND x∉B}A−B={x|x∈A AND x∉B}.
The complement of a set A (denoted by A′A′) is the set of elements which are not in set A. Hence, A′={x|x∉A}A′={x|x∉A}.
More specifically, A′=(U−A)A′=(U−A) where UU is a universal set which contains all objects.
Example − If A={x|x belongs to set of odd integers } A = {x|x belongs to set of odd integers} then A′={y|y does not
belong to set of odd integers} A′={y|y does not belong to set of odd integers}
The Cartesian product of n number of sets A1, A2,…AnA1,A2,…An denoted as A1×A2⋯×AnA1×A2⋯×An can be defined
as all possible ordered pairs (x1,x2,…xn)(x1,x2,…xn)where x1∈A1,x2∈A2,…xn∈Anx1∈A1,x2∈A2,…xn∈An
The following set properties are given here in preparation for the properties for addition and multiplication in
arithmetic.
Commutative Properties: The Commutative Property for Union and the Commutative Property for Intersection say that
the order of the sets in which we do the operation does not change the result.
Example: Let A = {x : x is a whole number between 4 and 8} and B = {x : x is an even natural number less than 10}.
Associative Properties: The Associative Property for Union and the Associative Property for Intersection says that how
the sets are grouped does not change the result.
Identity Property for Union: The Identity Property for Union says that the union of a set and the empty set is the set, i.e.,
union of a set with the empty set includes all the members of the set.
General Property: A ∪ ∅ = ∅ ∪ A = A
Example: Let A = {3, 7, 11} and B = {x : x is a natural number less than 0}.
Then A ∪ B = {3, 7, 11} ∪ { } = {3, 7, 11}.
The empty set is the identity element for the union of sets. What would be the identity element for the addition of
whole numbers? What would be the identity element for multiplication of whole numbers?
Intersection Property of the Empty Set: The Intersection Property of the Empty Set says that any set intersected with the
empty set gives the empty set.
General Property: A ∩ ∅ = ∅ ∩ A = ∅.
Example: Let A = {3, 7, 11} and B = {x : x is a natural number less than 0}.
Then A ∩ B = {3, 7, 11} ∩ { } = { }.
What number has a similar property when multiplying whole numbers? What is the corresponding property for
multiplication of whole numbers?
Distributive Properties: The Distributive Property of Union over Intersection and the Distributive Property of Intersection
over Union show two ways of finding results for certain problems mixing the set operations of union and intersection.
Example: Let A = {a, n, t}, B = {t, a, p}, and C = {s, a, p}. Then