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UNIT––2
SETS AND REAL NUMBERS
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CONTENTS
Introduction Objectives
1. Sets and Venn Diagrams
2. Types of Sets
3. Operations on Sets
4. Properties of Union and Intersection
5. Venn Diagrams and Verification of De Morgan’s laws
6. Real Numbers and their Properties
7. Radicals and Radicands
8. Laws of Exponents and Indices
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A mathematical set is a collection of distinct
objects, normally referred to as elements or
members.
2.1 What is Set
A set is a collection of distinct, symbols unordered objects. Sets are
typically collections of numbers, though a set may contain any type of data
(including other sets).The objects in a set are called the members of the set
or the elements of the set.
A set should satisfy the following:
1) The members of the set should be distinct. (not be repeated)
2) The members of the set should be well-defined. (Well-explained)
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Description of a Set
A= {2,4,6,8,…} B= {1,2, 3}
2,4,6, 8 are members or elements of the above set while it is called set A.
The curly brackets { } are sometimes called "set brackets" or
"braces". This is the notation for the two previous examples:
{socks, shoes, watches, shirts, ...}
{index, middle, ring, pinky}
The first set {socks, shoes, watches, shirts, ...} we call an infinite set,
the second set {index, middle, ring, pinky} we call a finite set. But
sometimes the "..." can be used in the middle to save writing long
lists: Example: the set of letters:
{a, b, c, ..., x, y, z}
In this case, it is a finite set because there are only 26 letters.
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The Number of Elements in a Set.
The number of elements in a set A, written as n[A],
is defined as the number of elements that A
contains.
For example,
if A = {a, b, c, d, e} , then n[A] = 5 (since
there are 5 elements in A); if D = (Sales,
Purchasing, Inventory, Payroll), then
n[D] = 4.
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Venn Diagrams
Example:
A = {1, 3, 5, 7,9}
Represent it by Venn diagram
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Subsets.
A subset of some set A, say, is a set which
contains some of the elements of
A. For example, if
A = {h,i,j,k,}), then:
X = {i, j, l} is a subset of A
Y = {h ,1} is a subset of A
Z = {i, ,b, j} is a subset of A and also a subset of
X
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2.2 Types of Sets
.
1.Finite set
A finite set has a limited number of members, such as the letters of
the alphabet.
Following are the examples of finite sets.
P = {Ali, Anam, Zara, Sara}
A= A set positive of multiples of 3 less than 32.
2. Infinite Set
An infinite set has an unlimited number of members,
P = {Ali, Anam, Zara, Sara, …}
Q= Set of all odd numbers.
D= Set of integers.
All these sets have unlimited numbers so all these are called infinite sets.
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3.Empty or Null set
An empty or null set has no members,
A= A set of Students having 4 legs.
B= A set of candidates hired as lecturers without masters degree.
C= A set of dogs with two heads
4. Singleton Set
A singleton set or single-element set has only one member,
A= Set of positive integers greater than 4 and less than 6.
B= Set of students securing gold medal in M.Sc Mathematics
from IUB in 2006 C={2}
D= Symbol of set of integers in Mathematics={z}
E= A set of Presidents of Islamic Republic of Pakistan
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5. Set equality/ Equal Sets
Two sets are equal only if they have identical elements. Or
we can say that Equal sets have the same members;
Example 1.
W = {days of the week} and
S = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}, both
the sets are equal. Or S=W
Example 2.
A= { 2, 4,6,8}
B= A set of first 4 positive multiples of 2. Here A = B
Example 3.
C= {1,2,3}
D= {1, 1+1, 2+1} C = D
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6. Equivalent sets
Sets with the same number of members are equivalent sets
Example 1.
A= {a, b, c }
B={1, 2, 3}
Here A is equivalent to B because number of elements are same in
both sets.
Example 2.
I = Months in Year
O = 1st twelve multiples of 2
I is equivalent to O because both sets have same number of elements
but keep in mind elements are not same just number of elements is
same.
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7.Overlapping sets
Sets with some members in common are intersecting sets;
if R = {red playing cards} and F = {face cards}, then R and
F share the members that are red face cards.
8. Disjoint Sets
Sets with no members in common are disjoint sets.; for example:
i. V = {vowels} and set L = {Set of alphabet started with letter l} are
disjoint sets because no vowels start with letter “l”.
ii. T= { Students of 10th
} and F={ Students of first year} is also disjoint
set because a student without passing tenth grade cannot get
admission in 1st
Year.
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9. The Universal Set
The set that contains all the elements or objects
involved in the problem under consideration. In some
problems involving sets, it is necessary to consider one
or more sets under consideration as belonging to some
larger set that contains them.
For example, if we are considering the set of all
students in a school then it will be the universal set. In
other words, where a universal set has been defined, all
the sets under consideration must necessarily be subsets
of it.
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2.3. Operations on Sets
2.3.1. Intersection of Sets
The intersection of two sets is the set containing the elements
common to the two sets and is denoted by the symbol ∩.
For example the intersection of A and B, denoted by A ∩ B, is the
set of all things which are members of both A and B. If A
∩ B = ∅, then A and B are said to be disjoint.
Examples: 1
{1, 2} ∩ {3,4}=∅.
{1, 2, 3} ∩ {3,4,5}= {3}.
{a, b} ∩ {a, b} = {a, b}.
15. A
U
B 3
2
7
8
1
9’ C
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Example: 2
For A={2,3,5,7,11}, B= {2,5,8} and C= {5,9,11}
Then determine the intersection of these sets and also represent it by using venn diagram..
Solution:
A∩(B∩C)= ({2,3,5,7,11}∩{2,5,8}) ∩{5,9,11}
1
= {2,5}∩{5,9,11} ={5}
(A∩B)∩C= {2,3,5,7,11}∩({2,5,8} ∩{5,9,11})
={2,3,5,7,11}∩{5}={5}
Verification by venn diagram
Shaded portion shown A∩ (B∩C) = 5 While from this figure we
can see that A∩B + {2, 5}
A∩C = {5, 11}
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Union of Sets
The union of two sets is the set containing all the elements of both sets
and is denoted by the symbol U. In other words we can say that union is
essentially the act of 'adding'
multiple sets together to combine their elements into a single set in such a
way that no element is
repeated.
Example. 1
If A= {1,3,5}
and B= {2,4,6}
Then the union of A and B is:
A𝖴B={1,3,5}U{2,4,6}= {1,2,3,4,5,6}
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Example: 2
For A={2,3,5,7,11}, B= {2,5,8} and C= {5,9,11}
Then determine the union of these sets and also represent it by using venn diagram..
Solution:
AU(BUC)= ({2,3,5,7,11}U{2,5,8}) U{5,9,11}
= {2, 3, 5, 7, 8, 11}U{5,9,11}
={2, 3, 5,7, 8, 9, 11}
(A𝖴 B) 𝖴 C= {2,3,5,7,11}∩({2,5,8} ∩{5,9,11})
={2,3,5,7,11}∩{2, 5, 8, 9, 11}
={2, 3, 5, 7, 8, 9, 11}
Verification by venn diagram
A
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A
B
Difference of Sets
The difference of two sets A and B (or relative complement of B in A), denoted
by A – B, is the set containing those elements that are in A, but not in B.
Example 1.
A = {1, 2, 3, 4, 5}
B= {3, 4, 5, 6, 7}
A-B = {1, 2, 3}
And
B-A = {6, 7}
Example 2.
A= {1, 3, 5}
B= {1, 2, 3}
Then A difference B will be:
A – B = {1, 3, 5} – {1, 2, 3} = {5}
B–A = {1, 2, 3} – {1, 3, 5} = {2}
Verification by Venn Diagram
Figure 2.3 By figure 2.3 lined portion represents AB = {5} Whiele
shaded region represents BA = {2}
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Complement of a set
The complement of a set A is all of the objects in
the universal set except those in A, or the
difference of universal set from any other set is
called the compliment of that set and is denoted
by A c
.
U = {1, 2, 3…10}
A = {1, 2, 3, 4}
A c
= U – A= {1, 2, 3…10} - {1, 2, 3, 4} = {5, 6, 7, 8, 9, 10}
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2.4. Properties of Intersection and Union
2.4.1 Some basic properties of
intersection are:
A ∩ B = B ∩ A.
A ∩ (B ∩ C) = (A ∩ B) ∩ C.
A ∩ B ⊆ A.
A ∩ B ⊆ B
A ∩ A = A.
A ∩ ∅ = ∅.
A ⊆ B if and only if A ∩ B = A.
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De, Morgan’s Laws
If A and B are the subsets of a universal set U, then
(AUB)C
= A c
∩ B C
(A∩B)C
= A c
U B C
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A ∩ BC
={1, 3, 5,………………99} ∩{ 2,4,6,…............................ 100}
= { } (ii)
By (i) & (ii) it is proved that (AUB) = A c
∩ B
Similarly
We can prove the second law
A∩B = { 2, 4, 6,………………100}∩{1, 3, 5,….........................99}
= { }
(A∩B) C
= U-(A∩B)
= { 1, 2, 3, ……………………100} - { }
= U (iii) A = U- A
= { 1, 2, 3, ……………………100} - { 2, 4, 6,…………………….100}
= {1, 3, 5,………………99} B = U-B
={ 1, 2, 3, ……………………100} - {1, 3, 5,………………99}
= { 2,4,6,…………………….100}
A U B ={1, 3, 5,………………99} U { 2,4,6,….............................100}
= {1, 2, 3,………………….100}
= U (iv)
By (iii) & (iv) it is proved that (A∩B)C
= A c
U B C
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2.6 - Real Numbers and Their Properties
To understand the concept of real numbers and their properties
it is necessary to understand rational and irrational numbers
and their properties. And to differentiate among rational and
irrational numbers it is necessary to understand the types of
decimal fractions.
Decimal fractions are of three types:
i. Terminating Decimal Fractions
a)
b)
c)
5/2 = 2.5
1/8 = 0.125
3/20= 0.15.
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Recurring and Non-terminating Fractions
a) 5/11 = 0.4545…..
In above example the recurring digits are 4 and 5.
b) 1/3 = 0.3333…..
In this example the recurring digits is 3. c) 1/6 =
0.16666……
In this example the recurring digits is 6.
Non Recurring and Non Terminating Fractions
( a) 5/7 = 0.7142875…..
(b) 11/13 = 0.8461538……
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2.6.1- Rational Numbers
The term "rational" comes from the word "ratio," because the
rational numbers are the ones that can be written in the ratio form
i.e. a/b where 'a' and 'b' are integers. In other words, a number is
rational if we can write it as a fraction where the numerator and
denominator are both Integers. The fraction which have exact
solutions are called rational fractions. The set of rational numbers
is denoted by Q.
Examples:
a) 4/5 = 0.8
b) 7/2 =3.5
c) 13/130= 0.1
The terminating and recurring non-terminating decimal fractions are
rational numbers.
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2.6.2- Irrational Numbers
An irrational number is any real number that cannot be expressed as a ratio
a/b, where a and b are integers, with b non-zero, and is therefore not a
rational number. We can also say that an irrational number cannot be represented
as a simple fraction. Irrational numbers are those real numbers that cannot be
represented as terminating or repeating decimals. The set of irrational numbers is
denoted by Q.
Famous Irrational Numbers
Pi is a famous irrational number. People have
calculated Pi to over one million decimal places
and still there is no pattern. The first few digits
look like this:
3.1415926535897932384626433832795…..
The number e (Euler's Number) is another famous
irrational number. People have also calculated e to
lots of decimal places without any pattern showing.
The first few digits look like this:
2.7182818284590452353602874713527……
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Real Numbers
A real number is a value that represents a
quantity along a continuous line. The real
numbers are the numbers that can be written in
decimal notation. The set of real numbers includes all
integers, positive and negative; all fractions; and the
irrational numbers, those whose decimal expansions never
repeat. We can say that the set of rational numbers and the
set of real numbers.
irrational numbers together form
Mathematically we can write as:
R = Q U Q
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Properties of Real Numbers
1. Closure Property
For all real numbers a & b the sum a + b and the
product a . b are real numbers.
2. Associative Law
For all real numbers a, b & c,
a + (b + c) = (a + b) + c and a . (b . c) = (a . b) . c.
3. Commutative Law
For all real numbers a & b, a
+ b = b + a and a . b = b . a.
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4. Distributive Law
For all real numbers a, b & c,
a . (b + c) = a . b + a . c and (a + b) . c = a . c + b . c.
5. Identity Elements
There are real numbers 0 and 1 such that for all real
numbers a, a + 0 = a and 0 + a = a, and
a . 1 = a and 1 . a = a.
6. Inverse Elements
For each real number a, the equations a +
x = 0 and x + a = 0
have a solution x in the set of real numbers, called the additive inverse of a,
denoted by -a.
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A radical expression is a number where both the
divisor and the dividend is the same. A common
radical is a square root. The symbol for a square root is
√.
We can say that the radical is the symbol that
represents a square root and the radicand is the number
underneath the radical symbol.
Example:
Before we can simplify a radical expression, we
must know the important properties of radicals.
PRODUCT PROPERTY OF SQUARE ROOTS
For all real numbers a and b,
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SIMPLIFYING RADICALS
The idea here is to find a perfect square factor of the
radicand, write the radicand as a product, and then use
the product property to simplify.
Example 1:
Simplify.
9 is a perfect square, which is also a factor of 45.
Use the product property.
If the number under the radical has no perfect square factors, then it cannot be
simplified further. For instance the cannot be simplified further because the only factors of 17 or 17 and 1.
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VARIABLE EXPRESSIONS UNDER THE RADICAL SIGN
When you have variables under the radical sign, see if you can factor out a square.
Example 6:
Simplify.
Factor the radicand as the product of a and a squared expression.
Use the product property of square roots:
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If bases are same but the powers are different then
we will add the powers on the same base
x a
x b
= (x) a+b
Division law of exponent
= (x) a-b
Law of exponent power of power
(x a
) b
= x ab
(xy)a
= xa
y a
=
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Fractional law of exponent
Question 1: Solve the exponents 3 7
× 3 2
Solution:
= 3 (7+2)
= 3 9
(using Multiplication law)
Question 2: Solve the exponents 2 (-3)
× (-7) (-3)
Solution:
= (2 × (-7)) (-3)
= (-14) (-3)
(using Power of power law)
