Class 11 || Maths || Ch. 01. Sets

 Class 11 || Maths || Ch. 01. Sets

CHAPTER – 01

SETS

Set: - Any well defined collection of objects, which are different from each other, and which we can see or think of is called a set. In short we can say ‘a set is well defined collection of objects’. 

Members or Elements of the sets: - The object which belongs to a set is called its members or elements. 

We give below a few more examples of sets used particularly in mathematics, viz.

N : the set of all natural numbers

Z : the set of all integers

Q : the set of all rational numbers

R : the set of real numbers

Z+ : the set of positive integers

Q+ : the set of positive rational numbers, and

R+ : the set of positive real numbers.

The symbols for the special sets given above will be referred to throughout this text.

Vinjeet – 01

Recognize which of the following are sets and why?

01. The collection of all months of a year beginning with letter J. 

Sol. These are only January, June and July so that it is well defined hence it is set. 

02. The collection of all even integers. 

Sol. We can definitely say that the collection of even integer is 2, 4, 6, …….. so it is set. 

1) The collection of ten most talented writers of India. 

2) A team of eleven best crickets’ batsmen of the World. 

3) The collection of all teachers in you school.

4) The collection of all boys in your class. 

5) The collection of the questions in this chapter.

6) The collection of most dangerous animals of the world. 

7) The collection of Natural numbers less than 100.

Look this set V = { a, e, I, o, u } and now we can read it ‘a’ belongs to V, or ‘a’ is the element of V or   and ‘b’ does not belongs to V or ‘b’ is not the element of V or  , etc. 

Let N = { 1, 3, 5, 7, 9 } then insert the appropriate symbol   or   in the blank. 

 

Representation of a Set: - The way of presentation of sets is known as representation of sets. There are two ways of representation of sets. 

01. Roster or Tabular form

02. Set-builder form 

01. Roster or Tabular Form: - In this form the elements of the set are beings separated by commas and are enclosed within brakes   {  }.

For example – 

The sets of all even positive integers less then 7

Sol. 

It is written in roaster form like that

N = { 2, 4, 6 }

The set of the vowels in English alphabets:

V = { a, e, I, o, u }

The set of the odd natural numbers < 10 

 N = { 1, 3, 5, 7, 9 }

The set of all natural numbers.

N = { 1, 2, 3, ………..}

02. Set-builder Form: - In this form, we list the property or properties satisfied by the elements of the sets. It is also written in brakes. 

For example – 

In the set  V = { a, e, i, o, u } Denoting this set by V we write

V = { x : x is a vowel in English alphabets}

N = { 1, 3, 5, 7, 9 }

N = { x : x is an odd natural numbers < 10}

N = { x : x is an odd natural numbers 0<x< 10}

N = { 1, 2, 3, ………..}

N = { x : x is a natural number.}

Jeet – 01- Write the solution set of the equation   in roster form.

Sol. 

By solving the equation to get the solutions. 

           

In roster form

Factor of   = { -2, 1}

Jeet – 2 – The Set of all natural numbers which divide 42 in roster form.

Sol.

Set = { 1, 2, 3, 6, 7, 14, 21, 42 }

Jeet – 3 – Write the set  { x : x is a positive integer and x2 < 40} in the roster form 

Sol.

Set = { 1, 2, 3, 4, 5, 6 }

Vinjeet – 02

01. Write the following in roaster form.

01. A = { x : x is an integer and  }

02. B = { x : x is   N,  }

03. C = { x : x is a natural number less than 6}

04. D = { x : x is a two digits numbers such that sum of its digits is 8} 

05. E = { x : x is   N,  }

06. F = {x : x is a set of all letters in the word TRIGONOMETRY.} 

07. G = {x : x is a set of all letters in the BUTTER.} 

02. Write the following sets in the set builder form. 

01. A = { 1, 2, 3, 4, 5, 6 }

02. B = { 1, 2, 3, 6, 7, 14, 21, 42 }

03. C = {  }

04. D = {  }

05. E = {  }

06. E = {  }

07. E = {  }

The Empty Sets: - The set which does not contain any element is called the empty set or null set or void set. It is denoted by    {   } or  

Finite and Infinite Set: - An empty set or a set which consist of a definite number of elements is called finite otherwise it is Infinite set. 

As 

The set of the days of the week is finite.

The set of the names of the months is finite.

The set of the solution of the equation is finite.

The set of points on the lines is infinite. 

The set of the prime number is infinite

The set of the odd number is infinite. 

Vinjeet – 03

Which are finite or infinite sets? 

01.  

02.  

03.  

04.  

05.  

06. The set of the months of the year.

07. The set of the positive integer less than 100.

08. The set of lines which are parallel to the x-axis.

09. The set of letters in English alphabets. 

10. The set of numbers which are multiple of 5.

11.  

12.  

The Equal Sets: - If two sets have same elements then they are called the equal sets. 

Mind it a set does not change if one or more elements of the sets repeated as 

A =   and B =  

These are equal sets. 

Which are equal sets?

01. A =   and B =  

02. A =   and B =  

03. A =   and B =  

04. A =   and B =  

05. A =   and B =  

06. A =   

B =  

07. A =   and B =  

Selects the equal sets

01. A =  

02. B =  

03.  

04.  

05.  

06.  

07.  

08.  

09.  

10.  

Subsets: - A set A is said to be a subset of a set B if every element of A is also an element of B.

 Which means implies

We can write the definition of subset as follows:

 

It is read as “A is a subset of B if a is an element of A implies that a is also an element of B”.

We are agree to say that  is a subset of every set.

If a set A has only one element, we call it a singleton set. Thus,{ a } is a singleton set.

Let A and B be two sets. If A ⊂B and A ≠B , then A is called a proper subset of B and B is called superset of A. For example,

A = {1, 2, 3} is a proper subset of B = {1, 2, 3, 4}.

01. Consider the sets  , A = { 1, 3 }, B = {1, 5, 9}, C = {1, 3, 5, 7, 9}. Insert the symbol  or  between each of the following pair of sets:

(i)  . . . B (ii) A . . . B (iii) A . . . C (iv) B . . . C

Sol. (i)  as  is a subset of every set.

(ii) A  B as 3  A and 3  B

(iii) A  C as 1, 3  A also belongs to C

(iv) B  C as each element of B is also an element of C.

Example 10 Let A = { a, e, i, o, u} and B = { a, b, c, d}. Is A a subset of B ? No. (Why?). Is B a subset of A? No. (Why?)

Example 11 Let A, B and C be three sets. If A  B and B  C, is it true that A  C?. If not, give an example.

Solution No. Let A = {1}, B = {{1}, 2} and C = {{1}, 2, 3}. Here A  B as A = {1} and B  C. But A  C as 1  A and 1  C.

Note that an element of a set can never be a subset of itself.

Subsets of set of real numbers

As noted in Section 1.6, there are many important subsets of R. We give below the names of some of these subsets.

The set of natural numbers N = {1, 2, 3, 4, 5, . . .}

The set of integers Z = {. . ., –3, –2, –1, 0, 1, 2, 3, . . .}

The set of rational numbers Q = { x : x =p/q , p, q  Z and q  0}

which is read “ Q is the set of all numbers x such that x equals the quotient p /q , where p and q are integers and q is not zero”. Members of Q include –5 (which can be expressed as  –5/1) , 5/7,   (which can be expressed as 7/2) and –11/3 .

The set of irrational numbers, denoted by T, is composed of all other real numbers. Thus T = {x : x  R and x  Q} = R – Q., i.e., all real numbers that are not rational. Members of T include   ,   and π .

Some of the obvious relations among these subsets are:

N  Z  Q, Q  R, T  R, N  T.

              

1.6.2 Intervals as subsets of R Let a, b  R and a < b. Then the set of real numbers { y : a < y < b} is called an open interval and is denoted by (a, b). All the points between a and b belong to the open interval (a, b) but a, b themselves do not belong to this interval.

The interval which contains the end points also is called closed interval and is denoted by [ a, b ]. Thus [ a, b ] = {x : a  x  b} 

We can also have intervals closed at one end and open at the other, i.e., 

[ a, b ) = {x : a  x < b} is an open interval from a to b, including a but excluding b. 

( a, b ] = { x : a < x  b } is an open interval from a to b including b but excluding a. 

These notations provide an alternative way of designating the subsets of set of real numbers. For example , if A = (–3, 5) and B = [–7, 9], then A  B. The set [ 0,  ) defines the set of non-negative real numbers, while set ( – ∞, 0 ) defines the set of negative real numbers. The set ( –  ,0) describes the set of real numbers in relation to a line extending from –  to  . On real number line, various types of intervals described above as subsets of R, are shown in the Fig 

 

Here, we note that an interval contains infinitely many points.

For example, the set {x : x  R, –5 < x  7}, written in set-builder form, can be written in the form of interval as (–5, 7] and the interval [–3, 5) can be written in setbuilder form as {x : –3  x < 5}.

The number (b – a) is called the length of any of the intervals (a, b), [a, b], [a, b) or (a, b].

1.7 Power Set

Consider the set {1, 2}. Let us write down all the subsets of the set {1, 2}. We know that is a subset of every set . So, is a subset of {1, 2}. We see that {1} and { 2 }are also subsets of {1, 2}. Also, we know that every set is a subset of itself. So, { 1, 2 } is a subset of {1, 2}. Thus, the set { 1, 2 } has, in all, four

subsets, viz. , { 1 }, { 2 } and { 1, 2 }. The set of all these subsets is called the power set of { 1, 2 }.

Definition 5 The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.

Thus, as in above, if A = { 1, 2 }, then

P( A ) = { ,{ 1 }, { 2 }, { 1,2 }}

Also, note that n [ P (A) ] = 4 = 22

In general, if A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2m.

1.8 Universal Set

Usually, in a particular context, we have to deal with the elements and subsets of a basic set which is relevant to that particular context. For example, while studying the system of numbers, we are interested in the set of natural numbers and its subsets such as the set of all prime numbers, the set of all even numbers, and so forth. This basic set is called the “Universal Set”. The universal set is usually denoted by U, and all its subsets by the letters A, B, C, etc.

For example, for the set of all integers, the universal set can be the set of rational numbers or, for that matter, the set R of real numbers. For another example, in human population studies, the universal set consists of all the people in the world.

EXERCISE 1.3