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Discrete Mathematics - Introduction
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities. This tutorial explains the fundamental concepts of Sets, Relations and Functions, Mathematical Logic, Group theory, Counting Theory, Probability, Mathematical Induction and Recurrence Relations, Graph Theory, Trees and Boolean Algebra. Mathematics can be broadly classified into two categories − Continuous Mathematics − It is based upon continuous number line or the real numbers. It is characterized by the fact that between any two numbers, there are almost always an infinite set of numbers. For example, a function in continuous mathematics can be plotted in a smooth curve without breaks. Discrete Mathematics − It involves distinct values; i.e. between any two points, there are a countable number of points. For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, and can be presented as a complete list of those pairs.
Topics in Discrete Mathematics
Though there cannot be a definite number of branches of Discrete Mathematics, the following topics are almost always covered in any study regarding this matter − Sets, Relations and Functions Mathematical Logic Group theory Counting Theory Probability Mathematical Induction and Recurrence Relations Graph Theory Trees Boolean Algebra We will discuss each of these concepts in the subsequent chapters of this tutorial. German mathematician G. Cantor introduced the concept of sets. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines. In this chapter, we will cover the different aspects of Set Theory.
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
Representation of a Set
Sets can be represented in two ways − Roster or Tabular Form Set Builder Notation
Roster or Tabular Form
The set is represented by listing all the elements comprising it. The elements are enclosed within braces and separated by commas.
Example 1 − Set of vowels in English alphabet, A={a,e,i,o,u}A={a,e,i,o,u}
Example 2 − Set of odd numbers less than 10, B={1,3,5,7,9}B={1,3,5,7,9}
Set Builder Notation
The set is defined by specifying a property that elements of the set have in common. The set is described
as A={x:p(x)}A={x:p(x)}
Example 1 − The set {a,e,i,o,u}{a,e,i,o,u} is written as −
A={x:x is a vowel in English alphabet}A={x:x is a vowel in English alphabet}
Example 2 − The set {1,3,5,7,9}{1,3,5,7,9} is written as −
B={x:1≤x<10 and (x%2)≠0}B={x:1≤x<10 and (x%2)≠0}
Example: 3- {a,b,c,b,c,a,a,c,b}= {a,b,c} repeated element
If an element x is a member of any set S, it is denoted by x∈Sx∈S and if an element y is not a member
of set S, it is denoted by y∉Sy∉S.
Example − If S={1,1.2,1.7,2},1∈SS={1,1.2,1.7,2},1∈S but 1.5∉S1.5∉S
Assignment: Specify the property base on the last example in Set builder notation
A set which contains infinite number of elements is called an infinite set.
Example − S={x|x∈NS={x|x∈N and x>10}x>10}
Subset
A set X is a subset of set Y (Written as X⊆YX⊆Y) if every element of X is an element of set Y.
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.
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 − S={x|x∈NS={x|x∈N and 7<x<8}=∅7<x<8}=∅
Singleton Set or Unit Set
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.
Example − If A={1,2,6}A={1,2,6} and B={16,17,22}B={16,17,22}, they are equivalent as
cardinality of A is equal to the cardinality of B. i.e. |A|=|B|=3|A|=|B|=
Overlapping Set
Two sets that have at least one common element are called overlapping sets. In case of 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 Difference/ Relative Complement
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}.
Example −If A={10,11,12,13}A={10,11,12,13} and B={13,14,15}B={13,14,15},
then (A−B)={10,11,12}(A−B)={10,11,12} and (B−A)={14,15}(B−A)={14,15}. Here, we can
see (A−B)≠(B−A)(A−B)≠(B−A)
Complement of a Set
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 belongstosetofoddintegers}A={x|x belongstosetofoddintegers} then A′={y|
y doesnotbelongtosetofoddintegers}A′={y|y doesnotbelongtosetofoddintegers}
Cartesian Product / Cross Product
The Cartesian product of n number of sets A 1 ,A 2 ,…AnA1,A2,…An denoted as A 1 ×A 2 ⋯×AnA1×A2⋯
×An can be defined as all possible ordered pairs (x 1 ,x 2 ,…xn)(x1,x2,…xn) where x 1 ∈A 1 ,x 2 ∈A 2 ,…
xn∈Anx1∈A1,x2∈A2,…xn∈An
Example − If we take two sets A={a,b}A={a,b} and B={1,2}B={1,2},
The Cartesian product of A and B is written as − A×B={(a,1),(a,2),(b,1),(b,2)}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)}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 2 n2n. Power set is denoted as P(S)P(S).
Example −
For a set S={a,b,c,d}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}{a},{b},{c},{d}
Subsets with 2 elements − {a,b},{a,c},{a,d},{b,c},{b,d},{c,d}{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}{a,b,c},{a,b,d},{a,c,d},{b,c,d}
Subsets with 4 elements − {a,b,c,d}{a,b,c,d}
Hence, P(S)=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}}
{{∅},{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)|=2 4 =16|P(S)|=24=
Note − The power set of an empty set is also an empty set.
|P({∅})|=2 0 =1|P({∅})|=20=
Partitioning of a Set
Partition of a set, say S , is a collection of n disjoint subsets, say P 1 ,P 2 ,…PnP1,P2,…Pn that satisfies the
following three conditions −
PiPi does not contain the empty set.
[Pi≠{∅} for all 0<i≤n][Pi≠{∅} for all 0<i≤n]
The union of the subsets must equal the entire original set.
[P 1 ∪P 2 ∪⋯∪Pn=S][P1∪P2∪⋯∪Pn=S]
The intersection of any two distinct sets is empty.
[Pa∩Pb={∅}, for a≠b where n≥a,b≥0][Pa∩Pb={∅}, for a≠b where n≥a,b≥0]
