Set and function.pptx

SETS AND FUNCTIONS
S.L.O no 1.1.1

Objectives :
At the end of the lesson, students will be able to identify the sets denoted by
N,Z,W,O,E,P,C and by other symbols;

What is Number System
•A numeral system (or system of
numeration) is a writing system for
expressing numbers, that is, a mathematical
notation for representing numbers of a
given set, using digits or other symbols in a
consistent manner.

Number System
•Natural Number (N)
•Whole Numbers (W)
•Integers (Z)
•Odd Numbers (O)
•Even Numbers (E)
•Prime Numbers (P)
•Composite Numbers ©

Natural Numbers
•Natural numbers are a part of the number system
which includes all the positive integers from 1 till
infinity. It is an integer which is always greater
than zero(0).
•Example; { 1, 2, 3, 4, 5, … }

Integers
•Integers. Integers are like whole numbers, but
they also include negative numbers ... but still no
fractions allowed! So, integers can be negative
{−1, −2,−3, −4, ... }, positive {1, 2, 3, 4, ... }, or
zero {0}

Whole Numbers
•whole numbers are the basic counting numbers 0, 1, 2, 3, 4,
5, 6, … and so on. 17, 99, 267, 8107 and 999999999 are
examples of whole numbers. Whole numbers include
natural numbers that begin from 1 onwards. Whole
numbers include positive integers along with 0.

Even Numbers
•An even number is a number that can be divided into
two equal groups.
OR
A number which is divisible by 2 and generates a
remainder of 0 is called an even number.
•Even numbers end in 2, 4, 6, 8 and 0
•All the numbers ending with 0,2,4,6 and 8 are even
numbers. For example, numbers such as 14, 26, 32, 40
and 88 are even numbers.

Odd Numbers
•An odd number is a number that cannot be divided into
two equal groups.
•All the numbers ending with 1,3,5,7 and 9 are odd
numbers. For example, numbers such as 11, 23, 35, 47
etc. are odd numbers.

Prime Numbers
•a number that is divisible only by itself and 1
(e.g. 2, 3, 5, 7, 11).
OR
prime numbers are whole numbers greater than 1,
that have only two factors – 1 and the number itself.

Composite Numbers
•A number that is divisible by a number other than 1
and the number itself, is called a composite number.
OR
A number that is a multiple of at least two numbers
other than itself and 1.

Set and Function
S.L.O No 2.1
Topic : Operation on Sets

Objectives :
End of the lesson students able to know about set there types

Key Words
• Set
• Description
• Roaster method or Tabular method.
• Set builder method or Rule method
• Cardinality of set

What is a set?
The theory of sets was developed by
German
• mathematician Georg Cantor
(1845-1918).
 He firstencountered sets while
working on “problems on
trigonometric series”.
 Sets are used to define the concepts of relations and functions. The study of
geometry, sequences, probability, etc. requires the knowledge of sets.
 Studying sets helps us categorize information. It allows us to
make sense of a large amount of information by breaking it
down into smaller groups.
Georg Cantor
(1845-1918)

A group or collection of well-defined distinct objects is
called a set .
When do we say that a collection is
“well-defined”?
When do we say that an object belongs to
a group?
¤ Each object in a set is called a member
or an element of a set.

Definition: A set is any collection of objects specified in such a way that
we can determine whether a given object is or is not in the collection.
 In other words A set is a collection of objects.
 These objects are called elements or members of the set.
 The symbol for element is .
 The following points are noted while writing a set.
 Sets are usually denoted by capital letters A, B, S, etc.
 The elements of a set are usually denoted by small letters a, b,
t, u, etc
Examples:
 A = {a, b, d}
 B = {math, religion, literature, computer science}
 C = { }

Sets
 Other ways to denote sets
 N = {1, 2, 3, 4. . .}
(set of natural numbers)
 Z = {. . ., -3, -2, -1, 0, 1, 2, 3,. . .}
(set of integers)
 E = {0, 2, 4, 6. . .}
(set of even natural numbers)
 Sets can be well defined.
 A well defined set is a set whose contents are clearly determined. The
set defined as “colors” would not be well defined while “the set of colors in a
standard box of eight crayons” is well defined.

There are three methods used to indicate a set:
• 1. Description
• 2. Roaster method or Tabular method.
• 3. Set builder method or Rule method

Description
 Description means just that, words
describing what is included in a set.
For example, Set M is the set of months that
start with the letter J.

Roster Form
•All elements of the sets are listed , each element separated
by comma(,) and enclosed within brackets
•e.g Set C= {1,6,8,4}
• Set T ={Monday,Tuesdy,Wednesday,Thursday,
Friday,Saturday}
• Set k={a,e,i,o,u}

Rule method or set builder method
• All elements of set posses a common property
• e.g. set of natural numbers is represented by
• K= {x|x is a natural no}
•Here | stands for ‘such that’ ‘:’can be used in place of ‘|’
• Set T={y|y is a season of the year} Set H={x|x is blood type}

Cardianility of set
• Number of element in a set is called as cardianility of set.
No of elements in set n (A)
e.g Set A= {he,she, it,the, you} Here no. of elements are n
|A|=5
Singleton set containing only one elements
e.g Set A={3}

Describe the following sets using thespecified methods.
A. Write a verbal description for each of the following sets:
1. D = {1, 3, 5, 7, . . . }
2. E = {a, b, c, . . . , z}
3. F = {4, 8, 12, 16, . . . , 96}
Answer:
1. The set of odd numbers.
2. The set of small letters in the English alphabet.
3. The set of multiples of 4 between 0 and 100.
Home Work

Describe the following sets using the specified methods.
B. List the elements of the following sets:
1. M = {x|x > 7, x is an odd integer}
2. A = {x|7 < x < 8, x is a counting number}
3. T = {x|x is a city in Metro Manila}
4. H = {x|x is a counting number between 7 and 10}
Answers:
1. M = {9, 11, 13, 15, 17, . . . }
2. A = { } or 
3. T = {Manila, Caloocan, Las Piñas, . . . Pasig}
4. H = {8, 9}
Class Work

Describe the following sets using thespecified methods.
C. Write a rule for the following: 1. S = {a, e,
i, o, u}
2. E = {3, 6, 9, . . . , 30}
3. T = {Monday, Tuesday, Wednesday, . . . , Sunday}
Home Work

Key Words
1. Empty set
2. Finite set
3. Infinite set
4. Equal set
5. Equivalent set
6. Disjoint set
7. Overlapping set
8. Subset Universal set

Empty sets
• A set which does not contain any elements is called as Empty set or Null or Void
set. Denoted by  or { }
e.g. Set A= {set of months containing 32 days}
Here n (A)= 0; hence A is an empty set.
e.g. set H={no of cars with three wheels}
• Here n (H)= 0; hence it is an empty set.

Finite set
• Set which contains definite no of element.
• e.g. Set A= {1,2,3,4}
• Counting of elements is fixed.
Set B = { x|x is no of pages in a particular book}
Set T ={ y|y is no of seats in a bus}

Infinite set
• A set which contains indefinite numbers of elements.
Set A= { x|x is a of whole numbers}
Set B ={y|y is point on a line}

Equal sets
• Two sets kand R are called equal if they have equal numbers and
of similar types of elements.
• e.g. If k={1,3,4,5,6} R ={1,3,4,5,6} then both Set k and R are equal.
• We can write as Set K= Set R

Equivalent Set
• Equivalent sets have different elements but have the same
amount of elements. If we want to write that
two sets are equivalent, we would use the tilde (~) sign.
A set's cardinality is the number of elements in the set.
Therefore, if two sets have the same cardinality, they
are equivalent!
• OR
• Two sets A and B are said to be equivalent if they have the
same cardinality .
• i.e. n(A) = n(B)
• Example. A={1,2,3} , B={2,3,4} then
• A ~ B

Disjoint Set
 Sets with no common elements are
calleddisjoint
• If A ∩ B = Ø, then A and B aredisjoint

Overlapping Set
• Two sets A and B are said to be overlapping if they contain at least
one element in common.
• For example; • A = {a, b, c, d} B = {a, e, i, o, u}

Subset
• Sets which are the part of another set are called
subsets of the original set. For example, if
A={3,5,6,8} and B ={1,4,9}
then B is a subset of A it is
represented as B  A
• Every set is subset of itself i.e A
 A
Empty set is a subset of
every set. i.e A
.3
.5
.6.
.8
.1
.9
A
.4 B

Universal set
• The universal set is the set of all elements
pertinent to a given discussion
It is designated by the symbol U
e.g. Set T ={The deck of ordinary playing cards}.
Here each card is an element of universal set.
Set A= {All the face cards} Set B=
{numbered cards}
Set C= {Poker hands} each of these sets are Subset
of universal set T

ACTIVITY 1
Determine whether the following is a set ornot:
1. The collection of all Math teachers.
2. Tall students in Grade 9.
3. Rich people in the Pakistan.
4. Planets in the Solar System.
5. Beautiful followers.
6. People living on the moon.
7. The collection of all large numbers.
8. The set of all multiples of 5.
9. A group of good writers.
10. Nice people in your class.

Power Sets
 Given any set, we can form a set of all possible subsets.
 This set is called the power set.
 Notation: power set or set A denoted as P(A)
 Ex: Let A = {a}
 P(A) = {Ø, {a}}
 Let A = {a, b}
 P(A) = {Ø, {a}, {b}, {a, b}}
• Let B = {1, 2, 3}
P(B)={Ø,{1},{2},{3},{1,2},{1,3}
,{2,3},{1,2,3}}

Applications
1.A set having no element is empty set. ( yes/no)
2.A set having only one element is singleton set. (yes/no)
3.A set containing fixed no of elements.{ finite/
infinite set)
4. Two set having no common element. ( disjoint set
/complement set)

Set and Function
S.L.O No 1.1.2 , 1.1.3 and 1.3.1

Objectives :
• End of the lesson students student able to understand about
operations of set

Operation on Sets
• Union of sets
• Intersection of sets
• Difference of two sets
• symmetric difference
• Complement of a set

Union
Definition: Let A and Bbe sets. The union of thesets
A and B,denoted by A ∪ B, is the set:
Example: What is
Solution: {1,2,3,4,5} U
A B
{1,2,3} ∪ {3, 4, 5}?
Venn Diagram for A ∪ B

Intersection
Definition: The intersection of sets A and B, denoted by A ∩ B,is
Note if the intersection is empty, then A and Bare said to be disjoint.
Example: What is {1,2,3} ∩ {3,4,5} ?
Solution: {3}
Example: What is
{1,2,3} ∩ {4,5,6} ?
Solution: ∅
U
A B
Venn Diagram for A ∩B

Difference
Definition: Let A and Bbe sets. The difference of A and B, denoted
by A–B,is the set containing the elements of A that are not in B.
The difference of A and Bis also called the complement of Bwith respect to
A.
A –B= {x| x ∈A x ∉ B} = A - B
U
A
B
Venn Diagram for A− B

Symmetric Difference
• Definition: The symmetric difference of Aand B, denoted by
• A B is the set
• Example:
• U = {0,1,2,3,4,5,6,7,8,9,10}
A = {1,2,3,4,5} B={4,5,6,7,8}
A B ={1,2,3,4,5} {4,5,6,7,8}
Solution: {1,2,3,6,7,8}
Venn Diagram

Universal Set
 A universal set is the super set of all sets under consideration
and is denoted by U.
 Example: If we consider the sets A, B and C as the cricketers of India,
Australia and England respectively, then we can say that the universal set (U)
of these sets contains all the cricketers of the world.
 The union of two sets A and B is the set which contains all those elements
which are only in A, only in B and in both A and B, and this set is denoted by
“A ŭB”.

Complement
 of a set is the set of all elements notin
• theset.
– Written𝐴𝑐 𝑜𝑟 𝐴′
– Need a universe of elements to drawfrom.
– Set U is usually called the universalset.
– Ac = {x| x U - A}
– B’= U- B

Working procedure to find
complement

Complement example
• A = { 1, 2, 3, 4} and Universal set = U = { 1, 2, 3, 4, 5, 6, 7, 8} then find A’
• A’ = U-A= { 1, 2, 3, 4, 5, 6, 7, 8} - { 1, 2, 3, 4}
• A’ = {5, 6, 7, 8} answer

Set and Function
S.L.O No 1.2.1, 1.2.2 , 1.3.1 and 1.3.2

Objectives :
• End of the lesson students student able to understand about
properties of union and intersection and there vein diagram

Commutative Properties
•The Commutative Property for Union and the Commutative
Property for Intersection say that the order of the sets in which we
do the operation does not change the result.

Commutative property of
union and intersection

Commutative Property of Union
• Let A = {x : x is a whole number between 4 and 8} and B = {x : x is an even
natural number less than 10}.
• W.r.t Addition
• AUB=BUA
• A ∪ B = {5, 6, 7} ∪ {2, 4, 6, 8} = {2, 4, 5, 6, 7, 8}
• BUA= {2, 4, 6, 8} ∪ {5, 6, 7} = {2, 4, 5, 6, 7, 8}
• Hence prove L.H.S = R.H.S

Commutative Property of Intersection
• Let A = {x : x is a whole number between 4 and 8} and B = {x : x is an
even natural number less than 10}.
• A ∩ B = B ∩ A
• A ∩ B = {5, 6, 7} ∩ {2, 4, 6, 8} = {6}
• B ∩ A = {2, 4, 6, 8} ∩ {5, 6, 7} = {6}
• Hence prove L.H.S=R.H.S

Associative Properties
•The Associative Property for Union and the Associative
Property for Intersection says that how the sets are grouped
does not change the result.

Associative property of
union and intersection

Associative Property of union
• (A ∪ B) ∪ C = A ∪ (B ∪ C)
• Let A = {a, n, t}, B = {t, a, p}, and C = {s, a, p}.
• L.H.S (A ∪ B) ∪ C
(A ∪ B) = {a, n, t} ∪ {t, a, p} = {p, a, n, t}
(A ∪ B) ∪ C = {p, a, n, t} U {s, a, p} = {p, a, n, t, s}
• R.H.S A ∪ (B ∪ C)
(B ∪ C) = {t, a, p} U {s, a, p} = {t, a, p, s}
A ∪ (B ∪ C) = {a, n, t} U {t, a, p, s} = {p, a, n, t, s}
Hence prove L.H.S = R.H.S

Associative Property of Intersection
• (A ∩ B) ∩ C = A ∩ (B ∩ C)
• Let A = {a, n, t}, B = {t, a, p}, and C = {s, a, p}.
• L.H.S (A ∩ B) ∩ C
(A ∩ B) = {a, n, t} ∩ {t, a, p} = {a,t}
(A ∩ B) ∩ C = {p, a, n, t} ∩ {a, t} = {a}
• R.H.S A ∩ (B ∩ C)
(B ∩ C) = {t, a, p} ∩ {s, a, p} = {a, p }
A ∪ (B ∪ C) = {a, n, t} ∩ {a, p } = {a}
Hence prove L.H.S = R.H.S

Identity Property for Union
The Identity Property for Union says that the union
of a set and the empty
set is the set, i.e., union of a set with the empty set
includes all the members of the set.

Identity Property for
Union and Intersection

A ∪ ∅ = ∅ ∪ A = A
•Example:
Let A = {3, 7, 11} and { }
Then A ∪ ∅ = {3, 7, 11} ∪ { }.
A ∪ ∅ = {3, 7, 11}

Intersection Property of the Empty Set
• The Intersection Property of the Empty Set says that any set intersected with the empty
set gives the empty set.
A ∩ ∅ = ∅ ∩ A = ∅.
Example: Let A = {3, 7, 11} and B = {x : x is a natural number less than 0}.
Then A ∩ B = {3, 7, 11} ∩ { } = { }.
B ∩ A = { } ∩ { 3,7,11} = { }

Distributive Properties
•The Distributive Property of Union over
Intersection and the Distributive Property of
Intersection over Union show two ways of finding results
for certain problems mixing the set operations of union
and intersection.
•A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
•and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Distributive Properties

Distributive Property of Union over Intersection
• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• Let A = {a, n, t}, B = {t, a, p}, and C = {s, a, p}. Then
L.H.S A ∪ (B ∩ C)
(B ∩ C) = { t, a, p} ∩ { s, a, p} = {a, p}
A ∪ (B ∩ C)= {a, n, t} ∪ {a, p} = {a, n, t, p}
R.H.S (A ∪ B) ∩ (A ∪ C)
AUB= {a, n, t} U {t, a, p} = {a, n, t, p}
A ∪ C = {a, n, t} U {s, a, p} = { a, n, s, t, p}
(A ∪ B) ∩ (A ∪ C) = {a, n, t, p} ∩ { a, n, s, t, p} = {a, n, t, p}

Distributive Property of Intersection
over Union
• A ∩ (B U C) = (A ∩ B) U (A ∩ C)
• Let A = {a, n, t}, B = {t, a, p}, and C = {s, a, p}. Then
L.H.S A ∩ (B U C)
(B U C) = { t, a, p} ∩ { s, a, p} = {a, t, s, p}
A ∩ (B u C)= {a, n, t} ∩ {a, t, s, p}= {a, t }
R.H.S (A ∪ B) ∩ (A ∪ C)
A ∩ B= {a, n, t} ∩ {t, a, p} = {a, t}
A ∩ C = {a, n, t} ∩ {s, a, p} = { a }
(A ∩ B) U (A ∩ C) = {a, t } ∩ { a } = {a, t }

De Morgan’s law
• The complement of the union of two sets is equal to the
intersection of their complements and the complement of the
intersection of two sets is equal to the union of their
complements. These are called De Morgan’s laws.
• (i) (A U B)' = A' ∩ B' (which is a De Morgan's law of union).
• (ii) (A ∩ B)' = A' U B' (which is a De Morgan's law of
intersection).

Working procedure to find De
Morgan’s law

Examples on De Morgan’s law:
• If U = {j, k, l, m, n}, X = {j, k, m} and Y = {k, m, n}.
• Proof of De Morgan's law: (X ∩ Y)' = X' U Y'.
• Solution:
L.H.S (X ∩ Y)'
U - (X ∩ Y)
(X ∩ Y) = {j, k, m} ∩ {k, m, n} = {k, m}
(X ∩ Y)' = {j, k, l, m, n} - {k, m} = {j, l, n}
R.H.S X' U Y'.
• X‘ = U – X = {j, k, l, m, n} - {j, k, m} = { l, n }
• Y' = U – Y = {j, k, l, m, n} - {k, m, n} = {j, l}
X' ∪ Y' = {l, n} ∪ {j, l}
Therefore, X' ∪ Y' = {j, l, n}
(X ∩ Y)' = X' U Y'. Proved

Examples on De Morgan’s law:
• Let U = {1, 2, 3, 4, 5, 6, 7, 8}, P = {4, 5, 6} and Q = {5, 6, 8}
Show that (P ∪ Q)' = P' ∩ Q'.
Solution:
• L.H.S (P ∪ Q)'
(P ∪ Q)‘ = U - (P ∪ Q)
P ∪ Q = {4, 5, 6} ∪ {5, 6, 8} = {4, 5, 6, 8}
(P ∪ Q)' = {1, 2, 3, 4, 5, 6, 7, 8} - {4, 5, 6, 8} = {1, 2, 3, 7}
• R.H.S P' ∩ Q'
• P‘= U – P= {1, 2, 3, 4, 5, 6, 7, 8} - {4, 5, 6} = {1, 2, 3, 7, 8}
Q’ = U – Q = {1, 2, 3, 4, 5, 6, 7, 8} - {5, 6, 8} = {1, 2, 3, 4, 7}
P' ∩ Q' = {1, 2, 3, 7, 8} ∩ {1, 2, 3, 4, 7}
P' ∩ Q' = {1, 2, 3, 7}
• (P ∪ Q)' = P' ∩ Q'. Proved

Set and Function
S.L.O No, 1.4.1 and 1.5.1

Objectives :
• End of the lesson students student able to understand Cartesian
product and Binary Relation

Cartesain Product
 Ordered pairs - A list of elements in which the order is significant.
 Order is not significant for sets!
 Notation: use round brackets.
• (a, b)
{a,b} = {b,a}
(a,b) (b,a)
 Cartesian Product: Given two sets A and B, the set of
– all ordered pairs of the form (a , b) where a is any
– element of A and b any element of B, is called the
– Cartesian product of A and B.
 Denoted as A x B
• A x B = {(a,b) | a A and b B}
• Ex: Let A= {1,2,3}; B = {x,y}
– A x B = {(1,x),(1,y),(2,x),(2,y),(3,x),(3,y)}
– B x A ={(x,1),(y,1),(x,2),(y,2),(x,3),(y,3)}
– B x B = B2 = {(x,x),(x,y),(y,x),(y,y)}
• (1, 2) • (2, 1)

Binary Relation
• Let P and Q be two non- empty sets. A binary relation R is defined to be a subset of
P x Q from a set P to Q. If (a, b) ∈ R and R ⊆ P x Q then a is related to b by R i.e.,
aRb. If sets P and Q are equal, then we say R ⊆ P x P is a relation on P
• (i) Let A = {a, b, c}
• B = {r, s, t}
• Then R = {(a, r), (b, r), (b, t), (c, s)}
• is a relation from A to B.
•
• (ii) Let A = {1, 2, 3} and B = A
• R = {(1, 1), (2, 2), (3, 3)}
• is a relation (equal) on A.

Working procedure of Relations

Examples Of Relations
• Example1: If A has m elements and B has n elements. How many relations are there
from A to B and vice versa?
• Solution: There are m x n elements; hence there are 2m x n relations from A to A.
• Example2: If a set A = {1, 2}. Determine all relations from A to A.
• Solution: There are 22= 4 elements i.e., {(1, 2), (2, 1), (1, 1), (2, 2)} in A x A. So, there
are 24= 16 relations from A to A. i.e.
• {(1, 2), (2, 1), (1, 1), (2, 2)}, {(1, 2), (2, 1)}, {(1, 2), (1, 1)}, {(1, 2), (2, 2)},
• {(2, 1), (1, 1)},{(2,1), (2, 2)}, {(1, 1),(2, 2)},{(1, 2), (2, 1), (1, 1)},
• {(1, 2), (1, 1), (2, 2)}, {(2,1), (1, 1), (2, 2)}, {(1, 2), (2, 1), (2, 2)}, {(1, 2), (2, 1), (1, 1), (2,
2)} and ∅.

Set and function.pptx

Recommended

Recommended

More Related Content

Similar to Set and function.pptx

Similar to Set and function.pptx (20)

Recently uploaded

Recently uploaded (20)

Set and function.pptx