Class 10 – Set Theory

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In these days of conflict between ancient and modern studies, there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but is the oldest and the youngest _ G.H. Hardy.

 

Introduction:The concept of set was developed by German mathematician Georg Cantor (1845 – 1918). He first encountered sets while working on “problems on trigonometric series”. Here, we will discuss some basic definitions and operations involving sets.

Sets and their representations:
In everyday life, we often speak of collection of objects of a particular kind, such as a pack of cards, a crowd of people, a cricket team, etc. In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc. More specially, we examine the following collections.

i)               Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9.

ii)             The rivers of India.

iii)            The vowels in the English alphabet, namely, a, e, i, o, u.

iv)            Various kinds of triangles.

v)             Prime factors of 210, namely, 2, 3, 5 and 7.

vi)            The solution of the equation: , viz, 2 and 3.

We note that each of the above examples is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belong to a given collection or not. For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this collection.

A few more examples of sets used particularly in mathematics, are given below 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. Again the collection of five most renowned mathematicians of the world is not a well – defined, because the criterion for determining a mathematician as most renowned may vary from person to person. Thus, it is not a well-defined collection.

We shall say that a set is a well-defined collection of objects.

The following points may be noted:

i)               Objects, elements and members of a set are synonymous terms.

ii)             Sets are usually denoted by capital letters A, B, C, X, Y, Z etc.

iii)            The elements of a set are represented by small letters a, b, c, x, y, z etc.

If a is an element of a set A, we say that “a belongs to A” the Greek symbol (epsilon) is used to denote the phrase ‘belongs to’. Thus, we write a  A.

 If ‘b’ is not an element of a set A, we write  and read “ does not belong to A”.

Thus, in the set V of vowels in the English alphabet, .

In the set P of prime factors of 30, 3 P but 15  P.

There are two methods of representing a set:

i)               Roster or Tabular form and

ii)             Set-builder form or Rule method.

1.              Roster form: In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces.

Example: The set of all even positive integers less than 7 is described in roster form as  Some more examples of representing a set in roster form are given below: 

a)     The set of all natural numbers which divide 42 exactly is

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngIn roster form, the order in which the elements are listed is immaterial. Thus, the above set can also be represented as

b)     The set of all vowels in the English alphabet is

c)     The set of odd natural numbers is represented by. The dots tell us that the list of odd numbers continue indefinitely.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngIt may be noted that while writing the set in roster form an element is notgenerally repeated. i.e., all the elements are taken as distinct elements. For example, theset of letters forming the word ‘SCHOOL’ is or. Here, theorder of listing elements has no relevance.

2.              Set-builder form:

In set – builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set. For example, in the set, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property. Denoting this set by V,
we write V =

It may be observed that we describe the element of the set by using a symbol (any other symbol like the letters y, z etc. could be used) which is followed by a colon “ : ”. After the sign of colon, we write the characteristic property possessed by the elements of the set and then enclose the whole description within braces. The above description of the set V is read as “the set of all  such that  is a vowel of the English alphabet”. In this description the braces stand for “the set of all”, the colon stands for “such that”. For example, the set.

A = is read as “the set of all  such that  is a natural number and lies between 3 and 10. Hence, the numbers 4, 5, 6, 7, 8 and 9 are the elements of the set A.

If we denote the sets described in (a), (b) and (c) above in roster form by A, B, C, respectively, then A, B, C can also be represented in set – builder form as follows:

A =

B =

C =

Example: Write the solution set of the equation  in roster form.

Solution:The given equation can be written as .

Therefore, the solution set of the given equation can be written in roster form as .

Example:Write the set in the roster form.

Solution: The required numbers are 1, 2, 3, 4, 5, 6. So, the given set in the roster form is .

Example: Write the set in set – builder form.

Solution:We may write the set A as

                                   

Alternatively, we can write

                                   

Example: Write the set in the set – builder form.

Solution:We see that each member in the given set has the numerator one less than the denominator. Also, the numerator begin from 1 and do not exceed 6. Hence, in the set – builder form the given set is

                                   

Example:Match each of the set on the left described in the roster form with the same set on the right described in the set – builder form:

i)                   (a)

ii)                                              (b)

iii)                       (c)

iv)                                      (d)

Solution: Since in (d), there 9 letters in the word PRINCIPAL and two letters P and I are repeated, so i) matches (d). Similarly, ii) matches (c) as  implies . Also,

1, 2, 3, 6, 9, 18 are all divisors of 18 and so iii) matches (a). Finally,

Implies  and so iv) matches (b).

The empty set: Consider the set              

We can go to the school and count the number of students presently studying in class X in the school. Thus, the set A contains a finite number of elements.

We now write another set B as follows:

B =

We observe that a student cannot study simultaneously in both classes IX and X. Thus, the set B contains no element at all.

Definition:A set which does not contain any element is called the empty set or the null set or the void set.

According to this definition, B is an empty set while A is not an empty set. The empty set is denoted by the symbol or .

Examples:

i)               Let. Then A is the empty set, because there is no natural number between 1 and 2.

ii)             B =. Then B is the empty set because the equation  is not satisfied by any rational value of

iii)            C =. Then C is the empty set, because 2 is the only even prime number.

iv)            D = . Then D is the empty set, because the equation is not satisfied by any odd value of x.

Finite and infinite sets: Let ,      B = and

We observe that A contains 5 elements and B contains 6 elements. How many elements does C contain? As it is, we do not know the number of elements in C, but it is some natural number which may be quite a big number. The number of elements of a set S, we mean the number of distinct elements of the set and we denote it by n(S). If n(S) is a natural number, then S is non-empty finite set.

Consider the set of natural numbers. We see that the number of elements of this set is not finite since there are infinite number of natural numbers. We say that the set of natural numbers is an infinite set. The sets A, B and C given above are finite sets and

Definition: A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.

Consider some examples:

i)               Let W be the set of the days of the week. Then W is finite.

ii)             Let S be the set of solutions of the equation . Then S is finite.

iii)            Let G be the set of points on a line. Then G is infinite.

When we represent a set in the roster form, we write all the elements of the set within braces . It is not possible to write all the elements of an infinite set within braces because the number of elements of such a set is not finite. So, we represent some infinite set in roster form by writing a few elements which clearly indicate the structure of the set followed (or preceded) by three dots.

For example,  is the set of natural numbers, is the set of odd natural numbers, is the set of integers. All these sets are infinite.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngAll infinite sets cannot be described in the roster form. For example, the set of real numbers cannot be described in this form, because the elements of this set do not follow any particular pattern. 

Example: State which of the following sets are finite or infinite:

i)               { and (x – 1) (x –2) = 0}

ii)             { and  = 4}

iii)            { and  –1 = 0}

iv)            { and  is prime}

v)             { and  is odd}

Solution:

i)               Given set = {1, 2}. Hence, it is a finite set.

ii)             Given set = {2}. Hence, it is a finite set.

iii)            Given set =. Hence, it is a finite set.

iv)            The given set is the set of all prime numbers and since set of prime numbers is infinite. Hence, the given set is an infinite.

v)             Since there are infinite number of odd numbers. Hence, the given set is an infinite.

Equal sets:Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal. Clearly, the two sets have exactly the same elements.

Definition: Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write .

Examples:

(i) Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B.

(ii) Let A be the set of prime numbers less than 6 and P be the set of prime factors of 30. Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngA set does not change if one or more elements of the set are repeated. For example, the sets A = {1, 2, 3} and B = {2, 2, 1, 3, 3} are equal, since eachelement of A is in B and vice-versa. That is why we generally do not repeat any element in describing a set.

Example:Find the pairs of equal sets, if any, give reasons:

A = {0},                                                           B = {> 15 and x < 5},

C = { – 5 = 0},                                          D =    

E = { is an integral positive root of the equation}.

Solution: Since  does not belong to any of the sets B, C, D and E, it follows that,

Since  but none of the other sets are empty. Therefore Also  but Hence,

Since. Further,  Thus, the only pair of equal sets is C and E.

Example: Which of the following pairs of sets are equal? Justify your answer.

i)               X is the set of letters in “ALLOY” and B is the set of letters in “LOYAL”.

ii)             A =  and B = .

Solution: We have, X =. Then X and B are equal sets repetition of elements in a set does not change a set. Thus,

                                   

ii)

Subsets:Consider the sets, X = set of all students in your school, Y = set of all students in your class. We note that every element of Y is also an element of X, then we can say that Y is a subset of X. The fact that Y is subset of X is expressed in symbolic form as  The symbol ‘’ stands for ‘is a subset of’ or ‘is contained in’.

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

In other words, if whenever, then. It is often convenient to use the symbol “” which means implies. Using this symbol, we can write the definition of subset as follows:

  

We read the above statement as “A is a subset of B if  is an element of A implies that is also an element of B”. If A is not a subset of B, then we write

We may note that for A to be a subset of B all that is needed is that every element of A is in B. It is possible that every element of B may or may not be in A. If it so happens that every element of B is also in A, then we shall also have . In this case, A and B are the same sets so that we have where is a symbol for two way implications, and is usually read as if and only if (briefly written as “iff”). It follows from the above definition that every set A is a subset of itself, i.e., Since the empty set  has no elements, we agree to say that  is a subset of every set. We now consider some examples:

i)               The set Q of rational numbers is a subset of the set R of real numbers, and we write Q R.

ii)              If A is the set of all divisors of 56 and B is the set of all prime divisors of 56, then B is a subset of A and we write B A.

iii)             Let A = {1, 3, 5} and B = {is an odd natural number less than 6}. Then A B and B A and hence A = B.

iv)             Let A = {a, e, i, o, u} and B = {a, b, c, d}. Then A is not a subset of B, also B is not a subset of A.

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}.

Description: http://images.clipartpanda.com/bloc-clipart-note-md.png If a set A has only one element, we call it a singleton set. Thus, { a } is a singleton set.

Example:Consider the sets

Insert the symbol  between each of the following pair of sets:

i)         ii)         iii)       iv)

Solution: i)  is a subset of every set.

   ii)

   iii)  also belongs to C

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

Example: Let  and .

Is A a subset of B ? No. there are elements of A which are not in B.

Is B a subset of A? No. there are elements of A which are not in B.

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

Solution: No. Let . Here  

Description: http://images.clipartpanda.com/bloc-clipart-note-md.pngAn element of a set can never be a subset of itself.

Subsets of set of real numbers:There are many important subsets of R. We are given below the names of some of these subsets.

The set of natural numbers

The set of integers

The set of rational numbers

which is read as “ Q is the set of all numbers  such that  equals the quotient

where p and q are integers and q is not zero”. Members of Q include –5 (which can be expressed as ),   (which can be expressed as ) and .

The set of irrational numbers, denoted by Q’, is composed of all other real numbers. Thus , i.e., all real numbers that are not rational. Members of include

Some of the obvious relations among these subsets are:

Intervals as subsets of R: Let a,  and . Then the set of real numbers  is called an open interval and is denoted by . All the points between a and b belong to the open interval  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

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

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

(a, b ] = {  <  ≤ 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 (– ∞, ∞) 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 figure

Here, we note that an interval contains infinitely many points. For example, the set , written in set-builder form, can be written in the form of interval as  and the interval  can be written in set- builder form as

The number is called the length of any of the intervals

Power set: Consider the set  Let us write down all the subsets of the set we know that  is a subset of every set. So,  is a subset of . We see that {1} andare also subsets of {1, 2}. Also, we know that every set is a subset of itself. So,  is a subset of . Thus, the set  has in all, four subsets, viz, ,  and. The set of all these subsets is called the power set of .

Definition: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

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

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.

Venn Diagrams: Most of the relationships between sets can be represented by means of diagrams which are known as Venn diagrams. Venn diagrams are named after the English logician, John Venn (1834-1883). These diagrams consist of rectangles and closed curves usually circles. The universal set is represented usually by a rectangle and its subsets by circles. In Venn diagrams, the elements of the sets are written in their respective circles.

Example: In the above given figure is the universal set of which

is a subset.

Example: In the figure given below is the universal set of which

 and B = are subsets, also .

 

Description: http://www.aids.harvard.edu/img/news/spotlight/newsletter_layout/logo.gifDifference between proper subset and power set.

Proper subset

Power set

Example: Let A = {1, 2}

                     B = {1, 2, 3}

In this case A is a proper subset of B

Power set of a set is the collection of all subsets of a set. Given by formula

          where m is the number of elements of set A.

Example: if A ={a, b}

 i.e., 4 subsets =