This document introduces the concept of sets. A set is a collection of distinct objects called elements or members. Sets can be finite, containing a countable number of elements, or infinite. The number of elements in a finite set is its cardinality. Sets are represented by capital letters and elements by lowercase. Properties of sets include membership, equality, subsets, unions, intersections, and complements. Venn diagrams provide a visual representation of relationships between sets. Basic counting rules apply to the number of elements in unions and intersections of finite sets.
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
CBSE Class 10 Mathematics Set theory Topic
Set theory Topics discussed in this document:
Introduction
Sets and their representations
Two methods of representing a set:
Roster form
Set-builder form
The empty set
Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
Intervals as subsets of R
Power set
Universal Set
Venn Diagrams
More Topics under Class 10th Set theory (CBSE):
Introduction to sets
Operations on sets
Visit Edvie.com for more topics
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
CBSE Class 10 Mathematics Set theory Topic
Set theory Topics discussed in this document:
Introduction
Sets and their representations
Two methods of representing a set:
Roster form
Set-builder form
The empty set
Finite and infinite sets
Consider some examples
Equal sets
Subsets
Subsets of set of real numbers:
Intervals as subsets of R
Power set
Universal Set
Venn Diagrams
More Topics under Class 10th Set theory (CBSE):
Introduction to sets
Operations on sets
Visit Edvie.com for more topics
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Empowering the Data Analytics Ecosystem: A Laser Focus on Value
The data analytics ecosystem thrives when every component functions at its peak, unlocking the true potential of data. Here's a laser focus on key areas for an empowered ecosystem:
1. Democratize Access, Not Data:
Granular Access Controls: Provide users with self-service tools tailored to their specific needs, preventing data overload and misuse.
Data Catalogs: Implement robust data catalogs for easy discovery and understanding of available data sources.
2. Foster Collaboration with Clear Roles:
Data Mesh Architecture: Break down data silos by creating a distributed data ownership model with clear ownership and responsibilities.
Collaborative Workspaces: Utilize interactive platforms where data scientists, analysts, and domain experts can work seamlessly together.
3. Leverage Advanced Analytics Strategically:
AI-powered Automation: Automate repetitive tasks like data cleaning and feature engineering, freeing up data talent for higher-level analysis.
Right-Tool Selection: Strategically choose the most effective advanced analytics techniques (e.g., AI, ML) based on specific business problems.
4. Prioritize Data Quality with Automation:
Automated Data Validation: Implement automated data quality checks to identify and rectify errors at the source, minimizing downstream issues.
Data Lineage Tracking: Track the flow of data throughout the ecosystem, ensuring transparency and facilitating root cause analysis for errors.
5. Cultivate a Data-Driven Mindset:
Metrics-Driven Performance Management: Align KPIs and performance metrics with data-driven insights to ensure actionable decision making.
Data Storytelling Workshops: Equip stakeholders with the skills to translate complex data findings into compelling narratives that drive action.
Benefits of a Precise Ecosystem:
Sharpened Focus: Precise access and clear roles ensure everyone works with the most relevant data, maximizing efficiency.
Actionable Insights: Strategic analytics and automated quality checks lead to more reliable and actionable data insights.
Continuous Improvement: Data-driven performance management fosters a culture of learning and continuous improvement.
Sustainable Growth: Empowered by data, organizations can make informed decisions to drive sustainable growth and innovation.
By focusing on these precise actions, organizations can create an empowered data analytics ecosystem that delivers real value by driving data-driven decisions and maximizing the return on their data investment.
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
2. What is a Set?
A set is a well-defined collection of
distinct objects.
The objects in a set are called the
elements or members of the set.
Capital letters A,B,C,… usually
denote sets.
Lowercase letters a,b,c,… denote
the elements of a set.
3. Examples
The collection of the vowels in the word
“probability”.
The collection of real numbers that
satisfy the equation .
The collection of two-digit positive
integers divisible by 5.
The collection of great football players in
the National Football League.
The collection of intelligent members of
the United States Congress.
0
9
2
x
4. The Empty Set
The set with no elements.
Also called the null set.
Denoted by the symbol f.
Example: The set of real numbers x
that satisfy the equation
0
1
2
x
5. Finite and Infinite Sets
A finite set is one which can be
counted.
Example: The set of two-digit
positive integers has 90 elements.
An infinite set is one which cannot
be counted.
Example: The set of integer
multiples of the number 5.
6. The Cardinality of a Set
Notation: n(A)
For finite sets A, n(A) is the number
of elements of A.
For infinite sets A, write n(A)=∞.
7. Specifying a Set
List the elements explicitly, e.g.,
List the elements implicitly, e.g.,
Use set builder notation, e.g.,
, ,
C a o i
10,15,20,25,....,95
K
/ where and are integers and 0
Q x x p q p q q
8. The Universal Set
A set U that includes all of the
elements under consideration in a
particular discussion.
Depends on the context.
Examples: The set of Latin letters,
the set of natural numbers, the set
of points on a line.
9. The Membership Relation
Let A be a set and let x be some
object.
Notation:
Meaning: x is a member of A, or x
is an element of A, or x belongs to
A.
Negated by writing
Example: . , .
A
x
A
x
, , , ,
V a e i o u
V
e V
b
10. Equality of Sets
Two sets A and B are equal, denoted A=B,
if they have the same elements.
Otherwise, A≠B.
Example: The set A of odd positive
integers is not equal to the set B of prime
numbers.
Example: The set of odd integers between
4 and 8 is equal to the set of prime
numbers between 4 and 8.
11. Subsets
A is a subset of B if every element of A is
an element of B.
Notation:
For each set A,
For each set B,
A is proper subset of B if and
B
A
B
A
A
A
B
Ø
B
A
12. Unions
The union of two sets A and B is
The word “or” is inclusive.
or
A B x x A x B
13. Intersections
The intersection of A and B is
Example: Let A be the set of even
positive integers and B the set of prime
positive integers. Then
Definition: A and B are disjoint if
and
A B x x A x B
}
2
{
B
A
Ø
B
A
14. 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
15. Venn Diagrams
A
Set A represented as a disk inside a
rectangular region representing U.
U
20. Sets Formed by Two Sets
o
R1 R3
U
A B
R2
R4
c
B
A
R
1
B
A
R
2
B
A
R c
3
c
c
B
A
R
4
21. Two Basic Counting Rules
If A and B are finite sets,
1.
2.
See the preceding Venn diagram.
)
(
)
(
)
(
)
( B
A
n
B
n
A
n
B
A
n
)
(
)
(
)
( B
A
n
A
n
B
A
n c