Discrete Mathematics Notes

Introduction to discrete mathematics notes:

Discrete Mathematics deals with several selected topics in Mathematics that are essential to the study of many Computer Science areas. Since it is very difficult to cover all the topics, only two topics, namely “Mathematical Logic” and “Groups” have been introduced. These notes will be very much helpful to the students in certain practical applications related to Computer Science. In this article we shall discuss about discrete mathematics notes.

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Discrete mathematics notes:


Logical statement or Proposition:

A statement or a proposition is a sentence which is either true or false but not both.

A sentence which is both true and false simultaneously is not a statement, rather it is a paradox.

Example 1:

(a) Consider the following sentences:

(i) The earth is a planet.

(ii) Rose is a flower.

Truth value of a statement:

The truth of a statement is called its truth value. If a statement is true, we say that its truth value is TRUE or T and if it is false, we say that its truth value is FALSE or F.

Simple statements:

A statement is said to be simple if it cannot be broken into two or more statements. All the statements in (a) and (b) of Example 1 are simple statements.

Compound statements:

If a statement is the combination of two or more simple statements, then it is said to be a compound statement.

Conjunction:

If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement ‘p and q’ is called the conjunction of p and q and is written in the symbolic form as ‘p ? q’.

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Discrete mathematics notes problems:

Example 1:

(i) Show that ((~ p) ? (~ q)) ? p is a tautology.

Solution:

(i) Truth table for ((~ p) ? (~ q)) ? p

p          q         ~ p      ~ q      (~ p) ? (~ q)     ((~ p) ? (~ q))? p

T          T          F            F                      F                      T

T          F          F           T                      T                      T

F          T          T          F                      T                      T

F          F          T          T                      T                      T

The last column contains only T. Therefore ((~ p) ? (~ q)) ? p is a tautology.

Example 2:

Show that ((~ q) ? p) ? q is a contradiction.

Truth table for ((~ q) ? p) ? q

p          q          ~ q      (~ q) ? p          ((~ q) ? p) ? q

T          T          F          F                                  F

T          F          T          T                                  F

F          T          F          F                                  F

F          F          T          F                                  F

The last column contains only F. ? ((~ q) ? p) ? q is a contradiction.

Discrete Mathematics Sample

Introduction discrete mathematics sample

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic do not vary smoothly in this way, but have distinct, separated values Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Having problem with Graph Ordered Pairs keep reading my upcoming posts, i will try to help you.

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Discrete mathematics sample explanations:


Here will study about the discrete mathematics problems.

The discrete mathematics contains the set of topics. These topics are cover in Tautologies and Logical Equivalence

Sentential Functions and Sets are in logic and sets. Relation and functions,   Equivalence Relations Equivalence Classes. Natural numbers, division and factorization.  Division  ,Factorization ,

Greatest Common Divisor. these are the some of the discrete mathematics

Here we will see some of the samples in the discrete mathematics

Example for sentence

1.The sentence \1 + 2 = 3 and 2 + 2 = 4" is true.

2.The sentence \3 + 3 = 6 and _ is rational" is false.

Example for relations and functions

Definition. Let A and B be sets. By a relative R on A and B, we mean a subset of the Cartesian

Product a x b.

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Discrete mathematics sample problems:


Here we will learn about the discrete mathematics sample problems

The natural numbers

Example1:

The set of natural numbers is usually given by

N = {1, 2, 3 …}

Division

Examples

Division we divide 24 by 4

Solution:

24/4 =6

We divide 24 by 4 we get answer is 6

Factorization

(x² +9 ) to factorize the given problem

Solution:

The general form of the given equation is

(x² +a²) =(x+a) (x+a)

So. We factorize given problem

(x² +9 ) =(x+3) (x+3)

Greatest Common Divisor

Example:

12,4,36

We find the greatest common divisor of the given problems?

Solution:

We form the given problem

We divide by 4 all the numbers

12=2x2x 3

4=2x2 x 1

36=2x2x 9

We get the greatest common divisor of the given problems

Final answer is 2

Discrete Mathematics Relation

Introduction to discrete mathematics relation:

The discrete mathematics deals with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.

Writing in set notation, if a is some fixed value:

` |{f(x)|x=a}|=1`

However, when we consider the relation, we relax this constriction, and so a relation may map one value to more than one other value. Having problem with Systems of Equations Solver keep reading my upcoming posts, i will try to help you.


Properties of discrete mathematics relation:-


In the following properties of discrete mathematics relation:-

Reflexive
Symmetric
Transitive
Antisymmetric
Trichotomy
Reflexive

Relation of the equality, = is reflexive. Examine that for all numbers a = a.So "=" is reflexive.

Symmetric

Relation is symmetric the values a and b:  a R b implies b R a.These type of relation is symmetric.

Transitive

Relation is transitive of all values of a, b, c: a R b and b R c implies a R c.These type of relation is transitive.

Antisymmetric

A relation is antisymmetric for all values a and b: a R b and b R a implies that a=b.These type of relation is antisymmetric

Trichotomy

A relation satisfy the all values a and b it holds true that: xRy or yRx.The  two relation numbers a and b, it is true that whether a ≥ b or b ≥ a (both if a = b).These type of relation is trichotomy. Please express your views of this topic How to Find Volume of a Cone by commenting on blog.


Example problems for discrete mathematics relation:-


Problem 1:-

`Let A = {1, 2, 3, 4, 5} and R : A harrA :-= {(a, b) : a |b}. What barR and R^-1?`

Solution:-

`{(1, 1) , (1, 2) , (1, 3) , (1, 4) , (1, 5) , (2, 2) , (2, 4) , (3, 3) , (4, 4) , (5, 5)}`

`barR = {(2, 1) , (2, 3) , (2, 5) , (3, 1) , (3, 2) , (3, 4) , (3, 5) , (4, 1) , (4, 2) , (4, 3) , (4, 5) , (5, 1) , (5, 2) , (5, 3) , (5, 4)}`

`R^-1 = {(1, 1) , (2, 1) , (3, 1) , (4, 1) , (5, 1) , (2, 2) , (4, 2) , (3, 3) , (4, 4) , (5, 5)}`

Problem 2:-

For each of the following relations of  pair which satisfies the relation ,and another pair which doesn't say whether each relation is reflexive relation,symmetric relation,transitive relation or anti-symmetric relation form discrete mathematics
(a) The relation on {1,2,3,4,5} defined by {(a,b)| a-b is even}
(b) The relation on {1,2,3,4,5} defined by {(a,b)| a+b is even}
(c) The relation on P, the set of all people,defined by {(a,b) | a and b have a common ancestor}

Solution:-

(a) The pair (4,2) satisfies the relation,(2,1) doesn't includes other relation.
(b) (3,3) satifies the relation,(3,2) doesn't includes other relation.
(c) (Bart,Lisa) satisfies the relation,(Homer,Marge) doesn't includes other relation.