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Sets and Relations
JOHN VENN :
John Venn is remembered chiefly for his logical diagrams. Venn became critical of themethods used in diagrams in the nineteenth century, especially those of George Boole andAugustus deMorgan. So Venn wrote the book Symbolic Logic mostly to interpret and make his own personal corrections on Boole's work, but this was not the reason Venn became so famous. Venn wrote a paper entitled On the Diagrammatic and Mechanical Representation of Prepositions and Reasonings introducing diagrams known today as Venn diagrams. In Symbollic Logic, Venn further elaborated on these diagrams, which became the most important part of his books.Venn extended Boole's mathematical logic and is best known to mathematicians and logicians for his diagrammatic way of representing sets, and their unions and intersections.Venn continued to improve his method for illustrating propositions by exclusive and inclusive circles. Venn's diagrams were the most consequential part of his logic trilogy,rather than his attempt to clarify what he believed to be inconsistencies and ambiguities in
Boole's logic.Later, he realized his diagrams were not sufficiently general so he extended his method by proposing a series of circles dividing the plane into compartments so that each
successive circle would intersect all the compartments already existing. This idea was taken up and refined by Charles Dodgson who lived from 1832 to 1898. Dodgson's ideasled to the use of the closed compartment, or what is now known as the universal set.
‘‘A set is any collection of distinct and distinguishable objects of our intuition or thought.’’By the term ‘distinct’ we mean that no object is repeated. By the term‘distinguishable’ we mean that given an object, we can decide whether that object is in our
collection or not.A set is represented by listing all its elements between braces { } and by separating them from each other by commas (if there are more than one element).Sets are usually denoted by capital Letters of English alphabet while the elements are denoted in general, by small latters.If x is an element of a set A, we write x A (read as ‘x belongs to A’). If x is not an
element of A, we writex A (read as ‘x does not belong to A’). The symbolis called the
membership relation. Here are some examples :
Example 1: Let A = {1, 2, 5, 2, 3}. The elements of this collection are distinguishable but not distinct, hence A is not a set.
Example 2: Let B = collection of all vowels in English alphabets. Then B = {a, e, i, o, u}.Here elements of B are distinguishable as well as distinct. Hence B is a set.
Example 3: C = Collection of all intelligent persons of Delhi. Here elements are not distinguishable because if we select any person of Delhi, we cann’t say with certainty whether he belongs to C or not, as there is no standard scale for evaluation of intelligence.
Representation of a Set
(i) Tabular form or Roster Form
Under this method elements are enclosed in curly brackets after separating them by
commas.
Example:If A is a natural number less than 5
A = {1, 2, 3, 4}
(ii) Set builder method
Under this method, set may be represented with the help of certain property or properties
possessed by all the elements of that set.
A = {x | P(x)} or A = {x : P(x)}
This signifies, A is the set of element x, such that x has the property P.
Example:
The set A = {1, 2, 3, 4, 5} can be written as
A = {x | x N and x 5}
Notations for Sets of Numbers
1. The set of all natural numbers, or the set of all positive integers is represented by N.
2. Set of whole numbers is represented by W.
3. Set of all integers is represented by Z or I.
4. Set of rational numbers is represented by Q.
5. Set of irrational number is represented by QC or Q(
6. Set of real numbers is represented by R.
7. Set of complex numbers is by C.
4.1 Finite and Infinite Sets
Finite set
A set having finite (definite) number of elements is called a finite set.
Example:
(i) Let A = {1, 2, 3}. Here A is a finite set as it has 3 elements (finite number of
elements).
(ii) Let B = set of all odd positive Integers
= {1, 3, 5, 7, 9,...}
Here B is not a finite set.
Infinite Set
A set which is not a finite set is called an infinite set. Thus a set A is said to be an infinite
set if the number of elements of A is not finite.
Examples:
(i) Let N = set of all positive integers = {1, 2, 3, 4, ...}
Here N is not a finite set and hence it is an infinite set.
(ii) Let Z = set of all integers = {..., –4, –3, –2, –1, 0, 1, 2, 3, 4, ...}
Here Z is an infinite set.
(iii) Let Q = set of all rational numbers
Here Q is an infinite set.
(iv) Let R = set of all real numbers.
Here R is an infinite et.
Cardinal Number of a Finite Set
The number of elements in a finite set A is called the cardinal number of set A and is
denoted by n (A).
Example: Let A = {1, 3, 5}, then n (A) = 3.
4.2 Equivalent and Equal Sets
Equivalent Sets
Two finite sets A and B are said to be equivalent if they have the same cardinal number.
Thus sets A and B are equivalent iff n (A) = n(B)
If sets A and B are equivalent, we write A = B
Example: Let A = {1, 2, 3, 4, 5}, B = {a, e, i, o u}. Here n (A) = n(B) = 5. Therefore sets A
and B are equivalent.