This document provides information about sets and set theory. It defines what a set is, discusses famous contributors to set theory like Cantor, De Morgan, and Venn, and covers different set operations and relations. Key points include that a set is a collection of objects, Georg Cantor formalized the study of sets, John Venn developed Venn diagrams to represent set relationships pictorially, and common set operations include union, intersection, difference, and complement.
Sets is the first lesson in Mathematics 7. This lesson introduces the basic terms. For more presentations visit me on YouTube. https://www.youtube.com/channel/UCltDbhOXh6r9FyYE52rWzCQ/playlists?shelf_id=18&view_as=subscriber&sort=dd&view=50
Professional physicists quickly learn the power and value of mathematical representations, not only as calculational tools, but as ways to organize conceptual knowledge and reason about physical situations. Often, this is because we started out with enjoyment and success in math and were enthralled by the idea that this beautiful stuff could actually be used to describe the world. Many of our students (especially those in service courses) don't come to physics with this orientation about math. An analysis of epistemological resources and stances chosen by physics faculty and students suggests that including math in our classes in the way most comfortable and natural to us as physicists might not help our students learn to use math in science. A more productive approach might be to run the math "upside down" by beginning with first building a strong physical intuition and then helping students translate to rigorous math.
Sets is the first lesson in Mathematics 7. This lesson introduces the basic terms. For more presentations visit me on YouTube. https://www.youtube.com/channel/UCltDbhOXh6r9FyYE52rWzCQ/playlists?shelf_id=18&view_as=subscriber&sort=dd&view=50
Professional physicists quickly learn the power and value of mathematical representations, not only as calculational tools, but as ways to organize conceptual knowledge and reason about physical situations. Often, this is because we started out with enjoyment and success in math and were enthralled by the idea that this beautiful stuff could actually be used to describe the world. Many of our students (especially those in service courses) don't come to physics with this orientation about math. An analysis of epistemological resources and stances chosen by physics faculty and students suggests that including math in our classes in the way most comfortable and natural to us as physicists might not help our students learn to use math in science. A more productive approach might be to run the math "upside down" by beginning with first building a strong physical intuition and then helping students translate to rigorous math.
Quantitative Data AnalysisReliability Analysis (Cronbach Alpha) Common Method...2023240532
Quantitative data Analysis
Overview
Reliability Analysis (Cronbach Alpha)
Common Method Bias (Harman Single Factor Test)
Frequency Analysis (Demographic)
Descriptive Analysis
Adjusting primitives for graph : SHORT REPORT / NOTESSubhajit Sahu
Graph algorithms, like PageRank Compressed Sparse Row (CSR) is an adjacency-list based graph representation that is
Multiply with different modes (map)
1. Performance of sequential execution based vs OpenMP based vector multiply.
2. Comparing various launch configs for CUDA based vector multiply.
Sum with different storage types (reduce)
1. Performance of vector element sum using float vs bfloat16 as the storage type.
Sum with different modes (reduce)
1. Performance of sequential execution based vs OpenMP based vector element sum.
2. Performance of memcpy vs in-place based CUDA based vector element sum.
3. Comparing various launch configs for CUDA based vector element sum (memcpy).
4. Comparing various launch configs for CUDA based vector element sum (in-place).
Sum with in-place strategies of CUDA mode (reduce)
1. Comparing various launch configs for CUDA based vector element sum (in-place).
Show drafts
volume_up
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.
3. What is a set?
• A well- defined, unordered collection or
group of objects of any kind
– The set of students in the Math 11 section O
class is an example of a set
– The set of subjects in the BSB program
Question: Is the group of smart students in
the CMSC 56 class a set?
14 June 2011 3Math 11 College Algebra
4. Famous Set Contributors
Georg Ferdinand Ludwig
Philipp Cantor
(1845 – 1918)
German mathematician
who made the first
formal study on sets;
published main paper
on sets in 1874
14 June 2011 4Math 11 College Algebra
5. Georg Cantor
• set of integers had an equal number of
members as the set of even numbers,
squares, cubes, and roots to equations
• number of points in a line segment is equal
to the number of points in an infinite line, a
plane and all mathematical space
14 June 2011 5Math 11 College Algebra
6. Famous Set Contributors
Augustus De Morgan
(1806-1871)
Responsible for the
idea of a universal
set
14 June 2011 6Math 11 College Algebra
7. Augustus De Morgan
• first person to define and name
"mathematical induction"
• developed De Morgan's rule to determine
the convergence of a mathematical series
• Formal Logic, his most important work
14 June 2011 7Math 11 College Algebra
8. Famous Set Contributors
John Venn
(1834 – 1923)
Considered the universal
set as the field of vision
Responsible for the
pictorial representation of
sets, the “Venn Diagram”
14 June 2011 8Math 11 College Algebra
9. Two Kinds of Sets According
to Size
• Finite sets – sets whose elements can be
enumerated or exhausted
Example: The set of BS Architecture
students in UP Mindanao
• Infinite sets- sets whose elements cannot
be itemized
Example: The set of whole numbers in the
real number system
14 June 2011 9Math 11 College Algebra
10. Symbols for Sets
1. Naming a Set: Capital letters are used to
name sets
2. Describing a Set: Two methods can be used
to describe a set
a. Roster Method – listing the elements of the
set
b. Rule Method – defining the set by the
common characteristic/s of the elements of
the set
14 June 2011 10Math 11 College Algebra
11. Symbols for Sets
• Membership in a Set – The symbol ∈ means
that the element is a member of the set, while
∉means the element is not a member of the set
Example: Let A = {marker, ballpen, pencil}
We can say that marker ∈A, while paper ∉A
14 June 2011 11Math 11 College Algebra
12. Symbols for Sets
• Empty Set or Null Set – refers to sets with
no elements, denoted by ∅ or { }
• Universal Set – the set of all objects under
consideration. This is usually denoted by U.
14 June 2011 12Math 11 College Algebra
13. Set Operations
• When we have two sets A and B, the
operations enumerated below can be applied:
1. Union – the set of all elements in the
universal set U, which belong in A or B
(A∪B)
Example: A = {1,2,3,4,5} , B = {2,4,6,8}
A∪B = {1,2,3,4,5,6,8}
14 June 2011 13Math 11 College Algebra
14. Set Operations
2. Intersection – the set of all elements in the
universal set U, which belong in both A and
B. (A∩B)
Example: A∩B = {2,4}
3. Difference – the set of all elements in U
which belong in A, but not in B (A – B)
Example: A – B = {1,3,5}
B – A = {6,8}
14 June 2011 14Math 11 College Algebra
15. Set Operations
4. Complement – The set of all elements in
the universal set U which are not in a
certain set. (A’)
Example: U = {1,2,3,4,5,6,7,8,9,10}
A = {1,2,3,4,5}
A’ = {6,7,8,9,10}
14 June 2011 15Math 11 College Algebra
16. Examples:
• Given: U = {1,2,3,4,5,a,b,c,d,6,7,e}
A = {7,3,a,b}
B = {1,2,3,6,7,c,d}
Find:
1. A∪B 3. A – B 5. A’
2. A∩B 4. B – A 6. B’
14 June 2011 16Math 11 College Algebra
17. Venn Diagram
• A picture representation of set relationships,
developed by John Venn (1834 – 1923).
• Properties:
– Elements of the universal set U are
represented by points in a rectangular region
– Members of sets in U are represented by
points within closed regions.
14 June 2011 17Math 11 College Algebra
19. U U
A BAB
A – B A’
14 June 2011 19Math 11 College Algebra
20. Set Relations
• Let A and B be sets in some universal set
U. Then A and B can be related according
to the following:
1. A is a subset of B: (“A ⊆ B”) - every
element of A is in B
- subsequently, the null set is a member of
every set.
14 June 2011 20Math 11 College Algebra
21. Set Relations
2. A equals B: (“A = B”) -every element of A
is also the element of B; A and B have
“identical elements”
3. A is a proper subset of B:(“A ⊂ B”)
- A is a subset of B but A ≠ B
4. A and B are disjoint: (“A ∩ B = ∅”)
-A and B have no common elements.
14 June 2011 21Math 11 College Algebra
22. Set Relations
5. A and B are complements: (A ∪B = U,
A ∩ B = ∅) A' = B and B' = A
6. A and B are equivalent: ( “ A~B ”) - A has
the same cardinality as B
Cardinality, |A| – number of elements in a set
Note: If A and B are disjoint: |A∪B| = |A| + |B|
If not, |A∪B| = |A| + |B| - |A ∩ B|
14 June 2011 22Math 11 College Algebra
23. In the Language of Venn…
U U
A B
B A
A = B A ⊆ B
14 June 2011 23Math 11 College Algebra
24. U
U
A ∩ B = ∅
A ~ B
A
B
A 1
2 3
B
a
b c
14 June 2011 24Math 11 College Algebra
25. Exercises
1. Fill in the blanks with the correct term:
a. A ream of______
b. A school of _____
c. A pack of ______
1. Indicate the sets in another form:
a. A = {x/x is a course in UP Mindanao}
b. B = {Davao, Digos, Tagum, Panabo}
c. C = {red, blue, yellow}
14 June 2011 25Math 11 College Algebra
26. Exercises
3. Indicate the elements of each set.
a. E = {v/v Є N, 0 < v < 3}
b. G = {a/a is a color on the Philippine
Flag}
c. H = {z/z is a Math subject of a
Comm. Arts student}
14 June 2011 26Math 11 College Algebra
27. 4. From the data compiled by the University
Registrar, it has been learned that 200 students
have enrolled in Math, 180 in History, while
170 have enrolled in English. Forty- five have
enrolled in both History and English, 50 in Math
and English and 40 in both Math and History.
Fifteen students, meanwhile, enrolled in all
three subjects. Out of all 500 students, how
many enrolled...
a.) in Math only?
b.) in both Math and English but not History?
c.) in at least one of these courses?
d.) in none of these courses?
14 June 2011 27Math 11 College Algebra
Editor's Notes
Well-defined = i.e., for any given object, it must be unambiguous whether or not the object is an element of the set
For example, if a set contains all the chairs in a designated room, then any chair can be determined either to be in or not in the set.
There must be a universe
(A union B)’ = A’ intersection B’
On the Diagrammatic and Mechanical Representation of Prepositions and Reasonings – a paper introducing diagrams (now known as venn diagrams)
paper
Fishes
Cigarettes
A={BSAM,BSBIO,BSCS,BSFT,BSARCHI,BSCOMMARTS,BAENG,BAANTHRO,BSABE,MASTER MGT,DIPLOMA URP,MASTER URP}
B=
C=