About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
A power point presentation on the topic SETS of class XI Mathematics. it includes all the brief knowledge on sets like their intoduction, defination, types of sets with very intersting graphics n presentation.
Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfSets.pdf
This pdf tackles about the Mathematical Language and Symbols and the Variables and the Language of Sets.
This presentation contains definitions, tables, illustrations as well as examples.
I hope you'll find this helpful.
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
About sets , definition example, and some types of set. Explained the some operation of set like union of set and intersection of set with usual number example
Sets & Set Operation
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 11, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
A power point presentation on the topic SETS of class XI Mathematics. it includes all the brief knowledge on sets like their intoduction, defination, types of sets with very intersting graphics n presentation.
Discrete Mathematics - Sets. ... He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or description. Set theory forms the basis of several other fields of study like counting theory, relations, graph theory and finite state machines.
After going through this module, you are expected to:
• define well-defined sets and other terms associated to sets
• write a set in two different forms;
• determine the cardinality of a set;
• enumerate the different subsets of a set;
• distinguish finite from infinite sets; equal sets from equivalent sets
• determine the union, intersection of sets and the difference of two sets
Lesson2_MathematicalLanguageAndSymbols _Lesson 2.1_VariablesAndTheLanguageOfSets.pdf
This pdf tackles about the Mathematical Language and Symbols and the Variables and the Language of Sets.
This presentation contains definitions, tables, illustrations as well as examples.
I hope you'll find this helpful.
Sections Included:
1. Collection
2. Types of Collection
3. Sets
4. Commonly used Sets in Maths
5. Notation
6. Different Types of Sets
7. Venn Diagram
8. Operation on sets
9. Properties of Union of Sets
10. Properties of Intersection of Sets
11. Difference in Sets
12. Complement of Sets
13. Properties of Complement Sets
14. De Morgan’s Law
15. Inclusion Exclusion Principle
Know the basics on sets such as the methods of writing sets, the cardinality of a set, null and universal sets, equal and equivalents sets, and many more.
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It is a sample of an interview for a business english class for pre-intermediate and intermediate english students with emphasis on the speking ability.
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Implicitly or explicitly all competing businesses employ a strategy to select a mix
of marketing resources. Formulating such competitive strategies fundamentally
involves recognizing relationships between elements of the marketing mix (e.g.,
price and product quality), as well as assessing competitive and market conditions
(i.e., industry structure in the language of economics).
2. ➢A set is a collection of well defined distinct
objects.
➢ Objects in the collection are called elements
of the set.
Introduction to Sets
3. Examples - set
•The collection of all books in our
library is a set.
Each book is an element of the set.
• The collection of all students is a
set.
Each student is an element of the
set.
4. Examples - set
The collection of counting numbers is a set.
Each counting number is an element of the
set.
The collection of pens in your briefcase is a set.
Each pen in your briefcase is an element of
the set.
5. Sets Notation
The simplest set notation used to represent the
elements of a set is the curly brackets { }.
An example of a set is A = {a, b, c, d}.
Here the set is represented with a capital letter, and
its elements are denoted by small letters.
6. Example – roster method
A variation of the simple roster method uses the
ellipsis ( … ) when the pattern is obvious and the set
is large.
{1, 3, 5, 7, … , 9017} is the set of odd
counting numbers less than or equal to 9017.
We call a finite set
{1, 2, 3, 4,… } is the set of all counting
numbers. We call an infinite set.
7. Example – set builder notation
• {x | x < 6 and x is a counting number} is the set of
all counting numbers less than 6.
• Note this is the same set as {1,2,3,4,5}
8. Notation – if an element of
If x is an element of the set A, we write this as
x A. x A means x is not an element of A.
If A = {3, 17, 2 } then
3 A, 17 A, 2 A and 5 A.
If A = { x | x is a prime number } then
5 A, and 6 A.
9. Definition
The set with no elements is called the
empty set or void or the null set and is
designated with the symbol .
10. Examples – empty set
⚫ The set of even prime numbers
greater than 2 is the empty set.
⚫ The set {x | x < 3 and x > 5} is the
empty set.
11. Definition - subset
The set A is a subset of the set B if every
element of A is an element of B.
• If A is a subset of B we write
A B to designate that relationship.
• If A is not a subset of B we write
A B to designate that relationship.
12. Example - subset
The set A = {3, 5, 7} is not a subset of
set B = {1, 4, 5, 7, 9}
because 3 is an element of A but is not an
element of B.
The empty set is a subset of every set,
because every element of the empty set
is an element of every other set.
13. Example - subset
The set
A = {1, 2, 3, 4, 5, 6} is a subset of the set
B = {x | x < 7 and x is a counting number}
Notice also that B is a subset of A because every
element of B is an element of A.
14. Proper Subset
A is a proper subset of B, written as A B,
when A is a subset of B and x B and x
A (every element A is in B but there is
at least one element B not in A)
When A is a proper subset of B, A B
Eg:
The set of all men is a proper subset of the
set of all people.
{1, 3} {1, 2, 3, 4}
{1, 2, 3, 4} {1, 2, 3, 4}
15. Example - equality
The sets
A = {3, 4, 6} and B = {6, 3, 4} are
equal because A B and B A.
The definition of equality of sets shows that
the order in which elements are written
does not affect the set.
16. Example - equality
The sets A = {2, 5} and
B = {2, 5, 7} are not equal
We would write A ≠ B.
Note that A B.
17. Definition - intersection
The intersection of two sets A and B is
the set containing those elements which
are elements of A and elements of B.
We write A B
18. If A = {3, 4, 6} and
B = { 1, 2, 3, 4, 5, 6}
then
A B = {3, 4, 6}.
Example - Intersection
19. Definition - union
The union of two sets A and B is the set
containing those elements which are
elements of A or elements of B without
duplication
We write A B
20. Example - Union
If A = {3, 4, 6} and
B = { 1, 2, 3, 5, 6} then
A B = {1, 2, 3, 4, 5, 6}.
24. Algebraic Properties
A few other elementary properties of
intersection and union.
A =A A ∩ =
A A = A A ∩ A = A
25. Universal Sets
The universal set is the set of all
things pertinent to a given discussion
and is designated by the symbol U
Example:
U = {all students at SPUIC}
Some Subsets:
A = {all IAB students}
B = {freshmen students}
C = {sophomore students}
26. The Complement of a Set
A
The shaded region represents the
complement of the set A
Ac
27. Set Complement
~A or
A′
“A complement,” or “not A” is the set
of all elements not in A.
*What the others have that you don’t*
28. Complements
o If A is a subset of the universal set U,
then the complement of A is the set
o Note: ;
=
c
A
A
c
A x U x A
=
U
A
A c
=
29. Cardinality of Sets
If A is a finite set, then cardinality of A,
denoted by n(A) or |A|, the number of
elements A contains
Eg: A = {x, y, z}. n(A) = 3
We see that
A B |A| |B|, and
A B |A| |B|.
30. Example
For U = {1, 2, 3, 4, 5}, A = {1, 2},
and B = {1, 2}, we see that A is a subset of
B (that is, A B), but it is not a proper
subset of B (or, A B).
A B |A| |B|, 2 2 so it is true that A
B
A B |A| |B|, 2 2 false so it is false
that A B.
31. Disjoint Set
Two sets are disjoint, if they have no elements in
common
A and B are disjoint, A B =
Eg: Are A = {1, 3, 5} and B = {2, 4, 6} disjoint?
Yes. {1, 3, 5} {2, 4, 6} =
Sets A1, A2, …, An are mutually disjoint, if no two sets Ai
and Aj with distinct subscripts have any elements in
common
Ai Aj = whenever i j
Eg: Are B1 = {2, 4, 6}, B2 = {3, 7}, B3 = {4, 5}
mutually disjoint?
No. B1 and B3 both contain 4.
32. Power Set
Power set of A, denoted by P(A), is the
set of all subsets of A
Theorem: If A B, then P(A) P(B)
Theorem: If set X has n elements, then
P(X) has 2n elements
Eg: P({x, y}) = {, {x}, {y}, {x, y}}
33. Power Set
Eg: Let C = {1, 2, 3, 4}. What is
the power set of C?
P(C) = {, {1}, {2}, {3}, {4}, {1,
2}, {1, 3}, {1, 4}, {2, 3}, {2, 4},
{3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3,
4},{2, 3, 4}, C}
34. Example
In a class of 50 college freshmen, 30 are studying C++,
25 are studying Java, and 10 are studying both
languages. How many freshmen are studying either
computer language?
We let U be the class of 50 freshmen, A is the subset of
those students studying C++, and B is the subset of
those studying Java.
n(A) = 30, n(B) = 25, n(A B) = 10
n(A B) = n(A) + n(B) - n(A B)
= 30 + 25 – 10
= 45
35. Example
Let U = {1, 2, 3, 4, 5, 6, x, y, {1, 2}, {1, 2, 3}, {1, 2, 3, 4}}
Then |U| = 11.
(a) If A = {1, 2, 3, 4} then |A| = 4 and
A U A U A U
{A} U {A} U {A} U
(b) Let B = {5, 6, x, y, A} = {5, 6, x, y, {1, 2, 3, 4}}. Then |B| =
5, not 8. And
A B {A} B {A} B
{A} B A B (that is, A is not a subset of B)
A B (that is, A is not a proper subset of B).
36. Based on the diagram, what is the total number of
students who did participate in volleyball?
15 9 11
6
5 4
12
Volleyball
Example