DISCRETE STRUCTURES THEORY:

                                                    MARWARI COLLEGE ,RANCHI
                                                     (Under Ranchi University ,Ranchi)

NAME : RAJU MANJHI
DEPT. OF CA ,MCR
SEM: B.Sc(CA)

SUB: DISCRETE STRUCTURES THEORY:

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Definition of Sets

A set is defined as a collection of distinct objects of the same type or class of objects. The purposes of a set are called elements or members of the set. An object can be numbers, alphabets, names, etc.

Examples of sets are:

a.             A set of rivers of India.

b.    A set of vowels.

We broadly denote a set by the capital letter A, B, C, etc. while the fundamentals of the set by small letter a, b, x, y, etc.

If A is a set, and a is one of the elements of A, then we denote it as a ∈ A. Here the symbol ∈ means -"Element of."

Sets  Representation:

Sets are represented in two forms:-

a) Roster or tabular form: In this form of representation we list all the elements of the set within braces { } and separate them by commas.

Example: If A= set of all odd numbers less then 10 then in the roster from it can be expressed as A={ 1,3,5,7,9}.

b) Set Builder form: In this form of representation we list the properties fulfilled by all the elements of the set. We note as {x: x satisfies properties P}. and read as 'the set of those entire x such that each x has properties P.'

Example: If B= {2, 4, 8, 16, 32}, then the set builder representation will be: B={x: x=2ⁿ, where n ∈ N and 1≤ n ≥5}

Cardinality of a Sets:

The total number of unique elements in the set is called the cardinality of the set. The cardinality of the countably infinite set is countably infinite.

Example:

1. Let P = {k, l, m, n}
The cardinality of the set P is 4.

2. Let A is the set of all non-negative even integers, i.e.
A = {0, 2, 4, 6, 8, 10......}.

As A is countably infinite set hence the cardinality.

Types of Sets

Sets can be classified into many categories. Some of which are finite, infinite, subset, universal, proper, power, singleton set, etc.

1.      Finite Sets: A set is said to be finite if it contains exactly n distinct element where n is a non-negative integer. Here, n is said to be "cardinality of sets." The cardinality of sets is denoted by|A|, # A, card (A) or n (A).

Cardinality of empty set θ is 0 and is denoted by |θ| = 0

Sets of even positive integer is not a finite set.

 Infinite Sets: A set which is not finite is called as Infinite Sets.

 Countable Infinite: If there is one to one correspondence between the elements in set and element in N. A countably infinite set is also known as Denumerable. A set that is either finite or denumerable is known as countable. A set which is not countable is known as Uncountable. The set of a non-negative even integer is countable Infinite.

 Uncountable Infinite: A set which is not countable is called Uncountable Infinite Set or non-denumerable set or simply Uncountable.

Example: Set R of all +ve real numbers less than 1 that can be represented by the decimal form 0. a₁,a₂,a₃..... Where a₁ is an integer such that 0 ≤ a_i ≤ 9.

3. Subsets: If every element in a set A is also an element of a set B, then A is called a subset of B. It can be denoted as A ⊆ B. Here B is called Superset of A.

Example: If A= {1, 2} and B= {4, 2, 1} the A is the subset of B or A ⊆ B.