Discreate Structure Brief your answer Write up to 7 sentence

Discreate Structure

Brief your answer. Write up to 7 sentences.

Solution

Discrete Mathematics on the other hand, involves distinct values; i.e. between any two
points, there are a countable number of points.

example, if we have a finite set ofobjects, the function can be defined as a list of ordered pairs having these objects, andcan be presented as a complete list of those pairs.

1.Sets, Relations and Functions:

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.

A set of all positive integers.

A set of all the planets in the solar system.

A set of all the planets in the solar system.

A set of all the planets in the solar system.

FUNCTION:

A function or mapping (Defined as f: XY) is a relationship from elements of one set X to
elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called
Codomain of function ‘f’.
Function ‘f’ is a relation on X and Y s.t for each x X, there exists a unique y Y such that
(x,y) R. x is called pre-image and y is called image of function f.
A function can be one to one, many to one (not one to many). A function f: AB is said
to be invertible if there exists a function g: BA

2.Mathematical Logic

Propositional Logic is concerned with statements to which the truth values, “true” and
“false”, can be assigned. The purpose is to analyze these statements either individually or
in a composite manner

example:

\"Man is Mortal\", it returns truth value “TRUE”

\"Man is Mortal\", it returns truth value “TRUE”

3.Group theory

4.Probability

Probability can be conceptualized as finding the chance of occurrence of an event.
Mathematically, it is the study of random processes and their outcomes. The laws of
probability have a wide applicability in a variety of fields like genetics, weather forecasting,
opinion polls, stock markets etc.

Probability of occurence of an event =Total number of favourable outcome/Total number of Outcomes.

Steps to find the probability:
Step 1: Calculate all possible outcomes of the experiment.
Step 2: Calculate the number of favorable outcomes of the experiment.
Step 3: Apply the corresponding probability formula.

5.Mathematical Induction and Recurrence Relations

Mathematical Induction is a mathematical technique which is used to prove astatement, a formula or a theorem is true for every natural number.
The technique involves two steps to prove a statement, as stated below:
Step 1(Base step): It proves that a statement is true for the initial value.
Step 2(Inductive step): It proves that if the statement is true for the nth iteration (or
number n), then it is also true for (n+1)th iteration ( or number n+1).

6.Graph Theory

7.Boolean Algebra

Boolean algebra is algebra of logic. It deals with variables that can have two discretevalues, 0 (False) and 1 (True); and operations that have logical significance.

A Boolean function is a special kind of mathematical function f: Xn X of degree n,
where X = {0, 1} is a Boolean domain and n is a non-negative integer. It describes the
way how to derive Boolean output from Boolean inputs.
Example: Let, F(A, B) = A’B’. This is a function of degree 2 from the set of ordered pairs
of Boolean variables to the set {0, 1} where F(0, 0) = 1, F(0, 1) = 0, F(1, 0) = 0 and
F(1, 1) = 0
Boolean Expressions
A Boolean expression always produces a Boolean value. A Boolean expression is
composed of a combination of the Boolean constants (True or False), Boolean variables
and logical connectives. Each Boolean expression represents a Boolean function.
Example: AB’C is a Boolean expression.
Boolean Identities
Double Complement Law
~(~A) = A
Complement Law
A + ~A = 1 (OR Form)
A · ~A = 0 (AND Form)
Idempotent Law
A + A = A (OR Form)
A · A = A (AND Form)
Identity Law

A + 0 = A (OR Form)
A · 1 = A (AND Form)