JEE Notes: Set Theory notes and important questions of JEE Mains and Advanced

JEE Notes of Set Theory, an important chapter for JEE Mains and Advanced.

JOHN  VENN :  
John Venn is remembered chiefly for his logical diagrams. Venn became critical of the methods used in diagrams in the nineteenth century, especially those of George Boole and Augustus de Morgan. 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 ideas led 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 write  (read as ‘x does not belong to A’). The symbol  is 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}

 

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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.

 

 

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.

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.

 

Equal Sets

Two sets A and B are said to be equal if each element of A is an element of B and each element of B is an element of A. Thus two set A and B are equal if they have exactly the same elements but the order in which the elements in the two sets have been written may differ. If sets A and B are equal, we write A = B.

Example:

         (i)  Let A = {1, 2, 3}, B = {1, 2, 3}

Here A and B have exactly the same elements.

Therefore A = B

(ii)  Let   A = {1, 2, 3, 4, 5, 6}

B = {x : x ∈ N and 1  x < 7}

Here A and B are equal sets.

Note:

  1. Two sets A and B are equal if
  2. Equal sets are equivalent sets but equivalent sets may or may not be equal.

Example:

Let A = {1,2, 3}, B = {a, b, c}

Here A and B have same number of elements and hence they are equivalent but they are not equal set as 1 ∈ A but 1 Î B.

Different Types of Sets

Null set (or empty set or void set)

A set having no element is called a null set or an empty set or void set. It is denoted by f or { }.

        Examples:

         (i)  Let A = the set of even numbers not divisible by 2.

(ii)  Let C = {x : x  N and 0 < x < 1}

Here C is a null set because there is no natural number lying between 0 and 1.

Note:     The set {0} is not an empty set a it contains are element 0.

Singleton Set

A set having single element is called a singleton set. It is represented by writing down the element within the braces.

        Examples:

(i)   A = The set of present prime minister of India.

(ii)  {2}, {0}, {ϕ}.

 

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Certain Useful Symbols

Symbols                                            Meaning

                                                         Implies

∈                                                         Belongs to

A ⊆ B                                                 A is a subset of B

                                                                         Implies and is implied by

Does not belong to

s.t. ( : or | )                                         Such that

”                                                         For every

There exists

iff                                                        If and only if

        a | b  a is divisor of b

 

ILLUSTRATIONS

Illustration 1

Represent the set A = {a, e, i, o, u} in set-builder form.

Solution

Let x denote an arbitrary element of A.

Then x can be any vowel of English alphabet.

  • A = {x : x is a vowel of English alphabet}.

Illustration 2

Write the set  in set-builder form.

Solution

We observe that in every element of set A, numerator is a natural number 1 to 6 and denominator is one more than the numerator.

Hence

Illustration 3

Represent the set A = {x : x is an odd integer and 3  x < 13} in tabular form.

Solution

Element of A is x which must have properties: It is an odd integer and is among                                        3, 4, 5, 6, 7,  8, 9, 10, 11, 12.

So they are 3, 5, 7, 9, 11.

    A = {3, 5, 7, 9, 11}.

Illustration 4

Match each of the sets on the left in the tabular (roster) form with the same set on the right given in the set-builder form :

(i) {5, 10, 15, ….}                            (a) {x : x is a letter of work COLLEGE}

(ii) {C, O, L, E, G}                           (b) {x : x is an odd natural number less than 10}

(iii) {1, 3, 5, 7, 9}                             (c) {x : x = 5n, n ∈ N}

(iv) {2, 3}                                          (d) {x : x is a prime number and a divisor of 12}

Solution

In (c), given set is the set of all natural numbers which are multiples of 5 and therefore (i) matches (c).

In (a), there are seven letters in the word COLLEGE and L and E occur twice, therefore (ii) matches (a).

In (b), given set is th set of all odd natural numbers less than 10 and therefore (iii) matches (b).

Prime divisors of 12 are 2 and 3 and hence (iv) matches (d).

 

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