Basic Set Theory

1.0 Introduction

A collection of well defined (distinct) objects is called a set (or a class) or a collection of objects under consideration, or is a collection of objects (items) which can be related in a particular manner. The objects can be a group of numbers, a group of persons, the books in a library, e.g. a set of silver ware, a set of ordered pairs, a set of solutions of equations, a set of types of cars, a set of dishes, a set of counting numbers, a set of integers and etc. The objects under consideration in a set are called the members or elements of the sets.

The books in a library constitute a set, the members or elements of this set being individual books in the library. The prime numbers 1,2,3,5,7,11, 13 …. Constitute a set and each prime number is an element (or member) of this set. A collection of all student girls in CBE-Dar es Salaam Campus is a set. But the collection of all ugly student girls in CBE-Dar es Salaam Campus isn’t a set, since the term ‘ugly’ is vague. Any one student girl may be classified as ‘ugly’ by one person and excluded by the other. A collection of all those boys in year one of Diploma whose weight is 62kg, is a set, since all its members (boys) can be ascertained for sure by a precise measurement of the weights of all the students in year one Diploma. Honest persons, beautiful women, bad boys, intelligent students, dull students, poor people, ugly men, aggressive people, coward persons, brave men, plump boys, interesting books, difficult problem and etc. do not form sets.

Note that the term ‘well defined’ means that it is defined in such a way that we are able to decide as to which object of the universe is there in our collections and which object is not there in our collections.

1.1Useful Symbols as Applied to Sets

In sets we have a number of useful symbols that are used in operating the sets or combining the sets together to bring a meaningful and clear understanding. These are:

{ } ‘braces’: they are used to enclose members or elements of a set in a ‘roster form’ or in a ‘listing method’

ϵ: is a symbol meaning ‘is   an element or member of’

 : is a symbol meaning ‘is not an element or member of’

Φ: meaning an empty, void or null set also can be shown as { }

 : A symbol meaning ‘is a proper subset of’

 : A symbol meaning ‘is a subset of’

 : meaning ‘is not a subset of’

U: it means a ‘Universal set’

 : meaning ‘Intersection’; it is a logical product(x); it means ‘AND’;’BOTH’ it’s called a ‘CAP’

 : meaning ‘Union’; it is a logical sum (+); it means ‘OR’; it is called a ‘CUP’

Ā, Á or   means ‘is a Complement of’, meaning ‘it doesn’t happen’ so other happens.

n (A): is a Cardinal number of a set A

: Or | is read ‘such that’; it is a ‘defining property method ’or ‘set-builder form’ of a set both can be used interchangeably.

I or Z are defined as ‘set of integers’ e.g. I= {….,-3, -2, -1, 0, 2, 3….}

N: indicates ‘standard sets of numbers’ e.g. N= {1, 2, 3, 4, 5….}

R or W: indicates ‘set of whole numbers’ e.g. W= {0, 1, 2, 3, 4, 5…}

1.2Sets notation

Usually sets can be written in capital letters, A , B, C, D, E etc. and their corresponding elements by small letters, a, b, c, d, e and etc.

If x is an element of set A, then we write xϵA, which means that ‘element x belongs to A’ or that “x is an element of A.

If x is not an element of set A, we say that “x doesn’t belong to A’ and this can be written as xϵA.

It is customary that the elements of a set are put curl brackets e.g. { }.

If A is the set of all even integers, then 2ϵ A but 7  A

If B is the collection of first six odd natural numbers, we write B = {1, 3, 5, 7, 9, 11}, clearly 1ϵ B, 3ϵ B, 5ϵB, 7ϵ B, 9ϵ B, 11ϵ B. We can say 2  B, 4  B, 6  B, 8  B, 10  B and etc.

X = {a, b, c, e} in indicates that X is the set with elements a, b, c, and e.

P = {1, 2, 3, 5, 7, 11…} indicates that P is the set of all prime numbers.

D = {{1, 2},{ 1,3}, {2,3}} shows that D is the set whose elements are sets {1,2}, {1,3}, and {2,3}.

A set may also be specified by the ‘defining property’ method or by ‘Rule’ method, or by ‘set builder form’. Under these methods, we state or list the requirements which an element must satisfy to belong to the set. (List the property or properties satisfied by elements of a set). The symbol ‘:’ or | (read “such that”) will be used as indicated here below: 

P= {x | x is a prime number} indicates that P is a set of all elements X such that X is a prime number i.e. P is the set of all prime numbers.

A = {x|x² -3x+2=0} indicates that A is the set of all elements X such that x² -3x +2=0, i.e. A = {1, 2}.

1.3Set representation

Sets may represented in three ways

As a description:  E.g. A= {all +ve odd numbers less than 10}, B = {all +ve even numbers}, C= {the result of throwing a die} and etc.

As a listing (roster method): E.g. A={1,3,5,7,9,…}, B={ 2,4,6,8 ,10 …}, C={1,2,3,4,5,6,…} and etc.

As a function (rule method): E.g.  A= {2x +1| 0<x>10}, B = {2x>0| xϵZ} here Z is the ‘set of integers”; C= {x| 0<x<7, xϵZ} and etc.

1.4Types of sets

When we talk of set types we can list seven types with some examples of each type.

Finite set: A set in which the process of counting of elements surely comes to an end (has countable number of elements) is called a finite set.

E.g. A set of all natural numbers less than 10, a set of integers lying between –ve 1000 and +ve 1000, a set of all vowels in the English alphabet having 5 elements namely a, e i ,o, u; a set of all persons on the earth, a set of all trees on earth and etc.

Infinite set:  A set that progresses without a limit is called an infinite set (process of counting of its elements does not come to an end).

E.g. Consider the set N={ 1,2,3,4,5,6,…} of all natural numbers(the process does not end) .Hence set N contains all natural numbers in the endless form.

W= {0, 1, 2, 3, 4, 5….} of all whole numbers; Z= {…, -3, -2, -1, 0, 1, 2, 3…..} of all integers;  The set A={…, -3, -2, -1, 0,1} of all integers less  than 2.

Each of the sets here under is infinite sets:

A set of all points on an arc of the circle; a set of all points on a line segment; a set of all concentric circles in a plane; a set of all natural numbers greater than 4000 etc.

Well- defined set: Let A= {2, 4 6, 8 10, 12…….} .This is a well-defined set because it is a set of even counting numbers since it begins with 2 and proceeds on increasing by 2

Empty set or Null set: This is a set having no elements; sometimes called void set and denoted by Φ. It is always represented is a roster method, as { }.

{X| xϵ N, 2 <X<3}=Φ. Clearly, there is no natural number lying between 2 and 3. Hence, the set of all natural numbers lying between 2 and 3 is an empty set.

{X |xϵ N, x ≠x}=Φ, clearly no natural number can be different from itself. Hence, the set of all those natural numbers each of which is not equal to itself is an empty set.

{X| x ϵ Z, X² = -1}=Φ, clearly, there is no integer whose square is -1. Hence, the set of all those integers whose square is -1 is a null set.

Note: The empty set is finite set since the number of elements contained in an empty set is zero.

Equal sets: When two sets A and B are said to be equal, then every elements of A is in B and every element of B is in A, and we write A=B.

If we let A ={x | xϵ N, x is a composite number, x < 9} and B ={x | x ϵ N, x is even, 3<x<9}. Then, A ={ 4,6,8} and  B ={ 4,6,8}. It is then true that every element of A is in B and every element of B is in A. Thus A=B.

Let M= set of all the letters in the word “PHOTOS’ and N= set of all letters in the word ‘HOTSPOT’. Then M= {P, H O T, S} and N= {H, O, T, S, P}. We see that, every element of M is in N and every element of N is in M. Therefore, M=N.

Note: (1) The elements of a set may be listed in any order. Thus, {a, b, c} = {b a, c} ={c, a, b} = {a, c, b}.

           (2) The repetition of elements in a set has no meaning. Thus, {1, 2, 1,2}= {1,2}; { 2, 4, 6, 2,6,4 } ={ 2,4,6} etc.

Clearly, if A, is the set of all letters in the word ‘CONCETRATE’ then we have A ={C, O, N, E, T, R, A}.

Equivalent sets: If two finite sets A and B are said to be equivalent, then they the same number of element, thus we write A‹—›B. Thus, whenever n (A) =n (B) then we have equivalent sets. We can clearly explain equivalent sets by use of Cardinal number of a set, i.e. the number of distinct elements contained in a finite set A. Cardinal number of a set is denoted by n(A).

If we let A = {2, 3, 5, 7, 11, 13,} then n (A) =6, and if B={ 2,4,6,8,10, 12} then n(B)=6 , clearly A‹—›B.

The null set Φ contains no elements at all so n (Φ) =0.

If we let P be the set of letters in the word ‘ JUNGLE’ and  Q be the set of letters in the word ‘KIDNAP’, then P={J, U, N ,G, L,E} and Q={K,I,D,N,A,P } thus n(P) =6 and n(Q)=6. Since n (P) =n (Q) =6 so P‹—›Q.

Note that two equal sets are always equivalent but equivalent sets need not be equal.

Let A={ a, b, c } and  B={ 1,2 3}, then n(A) =n(B)=3.Thus, A and B are equivalent. Clearly, A ≠ B

Disjoint sets: Two sets having no element in common are known as disjoint sets.

Let A= {2, 4, 6, 8} and B = {1, 7, 11}. It is very obvious that A and B have no elements in common. Hence A and B are said to be disjoint sets. 

The set of all odd natural numbers and the set of all even natural numbers are disjoint sets.

The set of all vowels in English alphabet and the set of all consonants in English alphabet are disjoint sets.

Overlapping sets: If two sets A and B have at least one element in common, then they are said to be overlapping sets.

If we let X to be the set of all vowels in English alphabet, and Y  be the set of 5 letters in English alphabet, then X={ a, e, i, o, u}, Y= {a, b, c, d ,e } are clear that X and Y have two elements in common namely a, e. So X and Y are overlapping sets.

Let A be the set of first five prime numbers, and B be the set of first five multiples of 2. Then A={2,3,5,7,11} and B={ 2,4,6,8,10}.It is obvious that A and B have one element in common namely 2. So set A and B are overlapping sets.

Exercises: 

1. Which of the following collections are sets?

(i) All books in your college library

(ii) All rose flowers in the college garden

(iii) All fiction movies

(iv) All honest people/persons in Dar es Salaam

(v) All student boys in diploma class weighing more than 50 kg.

(vi) All integers less than -5

2. Write each of the following sets in a roster form:

(i) A=set of all prime numbers between 60 and 80

(ii) C=set of all multiples of 3 between 15 and 30

(iii) E=set of all letters in the word ‘MATHEMATICS’

(iv) G=set of vowels in the word’ INTERMEDIATE’

(v) J= set of composite numbers between 10 and16

(vi) M= set of the months of the year starting with ‘J’

3. List the members contained in the following set, given that N is the set of positive integers and P is the set of all prime numbers.

(i)A={x: x ²=25}    (ii) H={x: 3x+2=0, or 2x+3=0} (iii) G= {x: x²-4x+3=0 and 2x²-3x+1=0} (iv) B={x: ϵ N and x is even} (v) F={x: x ϵ P and x is divisible by 3}

(vi) C= {x: x ϵ N and x έ P}.

4. Write each of the following sets in a builder form:

(i) A= {31, 37, 41, 43, 47} (ii) C= {7, 14, 21, 28, 35} (iii) D= {1, 2, 3, 6, 9, 18)

(iv) E={ 1, 4, 9, 16, 25, 36, 49}  (v) F={}  (vi) G={March, MAY}  (vii) J={½, 2/3, ¾,…8/9}

5. Write the Cardinal number of each of the following sets:

(i)A={x: x is a whole number, x<6} (ii) C ={x: x is a digit in the numeral, 2362}

(iii) E={x: x is an integer, -3<x<4}   (iv) G={x: x is a planet in the solar system}

(v) H={x: x ϵ W, and x-3<2} (vi) J = {x: x is a triangle having 4 sides)

6. In each of the following sets, state whether it is a finite of an infinite:

(i) A set of all rivers in China (ii) a set of all integers less than 7 (iii) a set of multiples of  (iv) a set of all points on a page (v) a set of drops of water in the bucket

(vi) A set of all even natural numbers

7. Identify the null sets (void sets, empty sets) among the given sets below:

(i)A={x: x is an even prime number}

(ii) B={x: x is an even number, x is not divisible by 2)

(iii)C={x: x is a whole number, x< 1}

(iv) D= {x: x is an integer, x²=-4}

(v) E={x: x is a perfect square number, 40< x <50}

(vi) F= {x: x is a number x>x}

8. State in each case whether the given pair of set consist of equal or equivalent but not equal sets or none:

(i)A= set of first five whole numbers, B= a set of first five natural numbers

(ii) Let P=set of letters of the word ‘FLOWER’, Q= set of letters of the word ‘FOLLOWER’ (iii) L= set of the letters of the word ‘OFFICER’ and M=set of letters of the word ‘PROFESSOR’

(iv) J={x: x ϵ 1, x+3=3} and K=Φ

(v)  H=set of all even numbers less than 12 and V=set all odd numbers less than 11

(vi)  A= set of equilaterals, B= set of all four sided closed figures

9. State in each case, whether the given sentences are T or F.

(i) If A is a set of all non-negative integers, then 0ϵA.

(ii) If C is the set of all prime numbers less than 80, then 57ϵ C.

(iii) Φ ϵ N    (iv) If E={x: x ϵ W, x <4} then n (E) =4 (v) {a, b, c, 1, 2, 3} is not a set.

(vi) {3, 5} ϵ {1, 3, 5, 7, 9}

10. From each of the following pairs of sets, identify the disjoint and overlapping sets.

(i)A={x: x is a prime number, x< 8} and B {x: x is an even natural number, x<8}

(ii) E={x: x is a person living in USA} and F={x: x is a person living in Washington, DC}

(iii) C= {x: x ϵ N, x <10} and D={x: x ϵ N, x is a multiple of 5}

(iv) G ={x: x is a multiple of 4} and H={ x: x is a multiple of 9}

(v) I={x: x is a perfect square, x < 30} and J={x: x is a factor of24}

(vi) K={x: x =8n, n ϵ N, and n< 7}

2.0 SUBSETS

 If A and B are two sets such that every element of A is in B, then we say that A is a subset of B and we write A  B ( is read ‘subset’). 

If there exists at least one element in A which is not a member of B, then we say that A is not a subset of B and we write A  B. Further subsets are divided into Superset and Proper subset.

Superset: Whenever a set A is subset of set B, we say set B is a superset of A and we write, B  A. Thus, whenever A B, then B A.

Let A= {2, 4 6, 8, 10} and B= {2, 4, 6, 8, 10}, then every element of A is in B. Hence, A B, and   B A

Let C be the set of all equilateral triangles and D be the set of all isosceles triangles. It is clear that since every equilateral triangle is an isosceles, so every element of C is a member of D. Therefore, C   D and hence D   C

Let A ={ 1,2,3,4} and B ={ 4,3,2,1}, then every member (element) of B is in A. We then say, B   A. In fact here A=B (equal sets). We generalize that whenever A=B then A   B and B   A.

Let A be any set. Then clearly every element of A is in A, therefore A   A. We then conclude that ‘every set is a subset of itself’.

An empty set (Φ) is a subset of every set, thus Φ   A, Φ   B and so on.

Proper subset:  If A and B are two sets such that A is a subset of B and A ≠ B, then we say that A is a Proper subset of B and  we write  A  B. Therefore, whenever every element of A is in B and n (A) < n (B) then we conclude that A   B.

Let A = {2, 3, 5} and B= {2, 3, 5, 7}. Then each member of A is in B and n (A) < n (B). Hence, A   B

Note: No set is a proper subset of itself. And the null set (Φ) is a proper subset of every set except itself.

2.1 Special Sets set 

Universal set: A set containing all elements (a set which is a superset of each one of the given sets). Such a set is called the Universal set, and is denoted by U or ξ. The Universal set changes from problem to problem.

Let A ={ 1,2,3,4,} , B = {2,3 5,7} and C ={2,4,6,8 } then  U ={1,2,3,4,5, 6,7,8} or any of its supersets may be  taken as Universal set.

Let D be the set of all odd natural numbers and E be the set of all even natural numbers. Then set N of all natural numbers may be taken as a Universal set.

The Null set: The set containing no members is known as Null set or Void or Empty set, denoted by { } or Φ.

Note that Φ and {Φ} are not the same. Φ is a Null set and {Φ} is a set containing an element Φ.

The Complement of a set (the opposite of a set): If we let U be the Universal set and let A   U, then the set of members of U which are not in A are called the Complement of A and is denoted by Á or Ā. Thus Ā ={x | x ϵ U and x έ A).

2.2Number of Subsets of a given set

Subsets can be formed by selecting 0, 1, 2, 3, 4, 5 ….n elements belonging to a given set. Hence the number of subsets will be equal 2ⁿ =ⁿC0 +ⁿC1 +ⁿC2 +……ⁿCn.  Alternatively, when we think of a subset being constructed by examining each of n-elements in turn and either refraining it or rejecting it. Then, there are 2x2x2x2x2….x2 =2ⁿ possible subsets. We therefore have a rule that ‘if a set is containing n- elements then it has 2ⁿ subsets.’

What are possible subsets of C= {a, b, c}? Here we see there are 7= (2³-1) proper subsets of C, i.e. Null set, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}.

Let U ={1,2,3,4,5,6,7} and let A= {2,4,6} .What is the complement of Ā?

We have Ā = set of those members of U which are not in A, then A= {1, 3, 5 7}.

What is the complement of Ā if we let U={x | x ϵ N, 70< X< 80} and A ={x | x is prime 70< x < 80}?

Here we have U = {71, 72, 73, 74, 75, 76, 77, 78, 79} and A = {71, 73, 79}. Ā =set of those elements of U which are not in A. Therefore A= {72, 74, 76, 77, 78} ={X | X is a composite number, 70< x < 80}.

Show that (i) U' =Φ (ii) Φ' =U (iii) Ā and A are disjoint.

By definition in (i) we have U' =U = set of those elements of U which are not in U =Φ. Therefore, Φ <=U'. In (ii) we have Φ' =set of those elements of U which are not in Φ=U    Φ' =U. And in (iii), by definition, Ā =set of those elements of U which are not in A. If we let x ϵ Ā, then x is an element of U not contained in A. So x is not contained in A. Hence, Ā and A are disjoint.

Exercises:

1.Let set A={ 2, 4, 6, 8, 10} ,set B={ 8, 10, 12}, C={ 2, 4, 8} and D={ 2, 10}. Which of the following statements are true?

(i) C A (ii) B A (iii) C D (iv) A B (v) B C (vi) D B

2. Let C be the set of all cyclic quadrilaterals, P=set of all parallelograms, Q= set of all quadrilaterals, R=set of all rectangles, S= set of all squares, T= set of all trapezia, V=set of all rhombuses. Now which of the following sub sets are correct?

(i) S R (ii) V P (iii) R P Q  (iv) T P (v) S R P T Q  (vi) T C Q.

3. Write down all the possible sub sets of each of the sets given here under:

(i) {2}, (ii) {3, 4} (iii) {1, 3, 5}, (iv) Φ (v) {c, d, e}, (vi) {2, 4, 6, 8} 

4. Write down all the possible proper sub sets of each of the sets given below:

(i) {c} (ii) {p, q} (iii){m, n, p} (iv) {1, 2, 3, 4} ( v) Φ

5. How many sub sets in all, are there of a set containing 4 elements?

6. How many sub sets in all, are there of a set with cardinal number 6?

7. Which of the following statements are true?

(i){a} {a, b, c}  (ii) {a} {b, c, d, e} (iii) Φ  { a, b, c}  (iv) Φ ϵ {a, b, c} (v) 0 ϵ Φ

(vi) Every sub set of finite set is finite 

8. Given that the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. Write down the sub sets of U containing (i) all odd numbers (ii) all prime numbers (iii) all numbers of multiples of 4.

9. Let set A= {x: x =2n-1, n< 6}. Find set A when:

(i) Universal set U=N, (ii) Universal set U=W, (iii) Universal set U=Z or I., 11}

10. Given that the universal U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, and let set A={2, 3, 5, 7, 11} , set B={ 2, 4, 6, 8,,10, 12} and set C= {1, 2, 5, 8, 10} be the sub sets of the universal set U. Find (i)Ā  (ii)complement B  (iii)complement C.

3.0Combination of Sets (Operations with Sets)

3.1Union of Sets: The symbol representing the Union of sets is ‘ ’.

If we let A and B represent two sets, then the union of the two sets A and B  is denoted by  A   B is the set consisting of all those members which are either in A or in B or both A and B. This relationship can also be shown in Venn-Diagrams in the next subsections of the Chapter.

In a defining set method this can be shown as A B ={x |x ϵ A or x ϵ B or both}. To list the elements of A B, we first write down all the elements of A and then take the elements of B which have not so far been taken up. There can be two or more sets consisting of all members in a set.

Note that A   B is read as ‘A plus B’ or ‘A OR B’ or ‘A cup B’ or ‘A union B’.

3.2Intersection of Sets: The symbol used in Intersections of sets is ‘ ’.

If we let A and B represent two sets, then the intersection of two sets A and B is denoted by A  B is the set consisting all those elements which are common  in both A and B . This relationship can also be depicted in a Venn-Diagram .Thus,  A   B = {x | x ϵ A and x ϵ B}. 

Note that A  B is read as ‘A times B’ or ‘A and B’ or ‘A cap B’ or ‘A intersection B. Note also that if x ϵ A or x ϵ B then x ϵ A  B, and if x ϵ A and x ϵ B then x ϵ A  B. The converse statements also hold.

3.3Difference of Sets: If we let A and B are both two sets then we can define them as:

A-B is the set of elements of A which are not in B and B-A is the set of elements of B which are not in A.

Suppose A={1,2,3,4,5 } and B={ 2, 4, 6, 8 } then A B ={1 ,2,3,4,5,6,7,8,} set consisting of all those members which are either in A or in B or both A and B. Also A B = {2, 4} the set of elements common to both A and B.

If we let A= {a, b, c, d}, B= {b, c, e}, C= {d, e, f} then we can verify that {i} A B =B A , and (ii) ( A B)  C = A  ( B  C).

Verification: We have in (i) A  B = {a, b, c, d}  {b, c, e} = {a, b, c, d, e},

                                      {B   A} = {b, c, e}  {a, b, c, d} = {b, c, e, a, d} = {a, b,       c, d, e}. Therefore A B =B  A.

In (ii) we have A  B= {a, b, c, d,}  {b, c, e} = {a, b, c, d, e}, and (A  B)  C ={a, b, c, d, e}   {d, e, f}={ a, b, c, d ,e, f}; again, B  C ={b, c, e}  { d, e, f} ={ b, c ,d ,e ,f}. Putting everything together, then A  (B  C) = {a, b, c, d} {b, c, d, e, f}={a, b, c, d, e, f}. Hence, (A  B)  C=A  (B C).

If A is the set of all dogs, and B is the set of all cats, then A B is the set of animals which are either dogs or cats and A B=Φ ( empty set, void set, null set).

If A is any set, then A Ā=U and A Ā=Φ (null set, void set, empty set).

Suppose M={x |x is the factor of 12} and N={X | X is the factor of 16}. Find M N?

Solution:  Here we have, M={ x| x is the factor of 12} =[ 1 ,2, 3, 4, 6,12} and N={ x | x is a factor of 16} ={ 1, 2, 4, 8, 16} .Therefore M N={1, 2, 3, 4, 6, 12} {1,2,4,8,16}={1,2,4}.We can as well find M N={1,2,3,4,6,12} {1,2,4,8,16}={1,2,3,4,6,8,12,16}.

If P={x |x is a positive integer, x<6} and Q={x | x is a negative integer x> -6}. Find P Q?

Solution:  We have P= {x| x is a positive integer, x<6} = {1, 2, 3, 4, 5}, and Q={x | x is a negative integer, x>-6} = -5, -4, -3, -2, -1}. Thus P Q= {1, 2, 3, 4, 5} {-5, -4, -3, -2, -1}=Φ( empty set, null set, void set). Note however that whenever such a situation happens as P Q =Φ, we always state the sets are disjoint sets.

Find A-B and B-A given that A={ 2,3,5,7,9} and B={ 2,3,4 5,6}.

Solution:  A-B is a set of elements of A which are not in B, i.e. B= {7, 9} and B-A is a set of elements of B which are not in A, i.e. A= {4, 6}.

Given that M= {3, 6, 9 12, 15} and N= {16, 12, 18, 24}. Find M-N and N-M.

Solution: We have: M-N is the set of elements of M which are not in N= {3, 9 15}, and N-M is the set of elements of N which are not in M = {18, 24}.

Exercises: 

1. Given that E is the set of all English speaking mathematicians, G is the set of all German speaking mathematicians, R is the set of all Russian speaking mathematicians, and U, is the universal set of all the mathematicians, describe the following sets in words: 

(i) E G (ii) R G (iii) R E' (iv) G' E (v) (E R)' (vi) (E' R') G.

2. The set of all subsets of a given set A is called the power set of A, and is written P (A). Show that the number of elements in P (A) =2ⁿ, where n is the number of elements is set A. If set A ={a, b, c} and set B ={ b, c, d) write out all the elements of sets P(A),P(B) and P(A B).

3. Given that set A= {x | x is a multiple of 2, x< 20}, set B= {x | x is a multiple of 3, x< 20), set C={x: x is prime, x< 20}, D={x: x is a perfect square natural number, x< 20}.

(i)Write each one of the sets A, B, C, D in a Roster form.

(ii) Find (a) A B (b) A C (c) A D (d) B C (e)B D (f) A B (g) A C (h) B D.

4. Let set A= {a, b, c}, set B= {b, d, e}, set C= {e, f, g}. Verify that:

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

5. Let U be the universal set = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and let set A= {2, 3, 4, 5, 6} and set B = {3, 5, 7, 8} be the sub sets of universal set U. Find:

(i)A B (ii) A B (iii) A-B (iv)B-A (v)A B' (vi) A' B (vii) A' B' (viii) A' B'.

6. Given that the universal set U is the set of all positive integers, A is the set of all positive integers less than 6 or equal to 6, E is the set of all even positive integers, M is the set of all positive integers which are multiples of 3. Find simple expressions in terms of sets A, E, and M for the following sets:

(i) {3, 6} (ii) {1, 3, 5} (iii) all positive integers which are multiples of 6

(iv) All even integers greater than 6 (v) the set which contains all multiples of 3 and all odd integers.

4.0Venn-Euler diagrams (commonly as Venn diagrams)

Venn-Euler diagrams are used to represent sets by specific regions. They express relationship among sets in a more significant way. They are pictorially represented by means of diagrams. Although Venn-Euler diagrams are typically used in case of two or three sets, there is no limit to the number of sets we could use.

The Universal Set is represented by a rectangle and its subsets by circles (or closed bounded figures) drawn inside this rectangle.