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This document contains lecture notes on fundamental concepts in discrete mathematics including: - The cardinality or size of a set is the number of elements it contains and is denoted by the symbol |S|. - The Cartesian product of two sets A and B contains all ordered pairs (a,b) where a is in A and b is in B. - Set operations like union, intersection, difference and complement are defined and explained with examples. - Venn diagrams are introduced to graphically represent relationships between sets. - Disjoint sets, generalized unions and intersections of multiple sets are also covered.

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Sets functions-sequences-exercises

This document provides an overview of basic discrete mathematical structures including sets, functions, sequences, sums, and matrices. It begins by defining a set as an unordered collection of elements and describes various ways to represent sets, such as listing elements or using set-builder notation. It then discusses operations on sets like unions, intersections, complements, and Cartesian products. Finally, it introduces functions as assignments of elements from one set to another. The document serves as an introduction to fundamental discrete structures used throughout mathematics.

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- There are different types of sets such as the empty/null set, singleton sets, finite sets, and infinite sets.
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- A set is a well-defined collection of objects or elements. Sets can be defined by listing elements or describing membership rules.
- Common notations are presented for defining sets, elements, membership, subsets, unions, intersections, complements, and cardinality.
- Finite and infinite sets are discussed. Special sets like the empty set, power set, and universal set are introduced.
- Venn diagrams are used to visually represent relationships between sets such as subsets, unions, intersections, and complements.

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- A set is a well-defined collection of objects. The empty set is a set with no elements. A set is finite if it has a definite number of elements, otherwise it is infinite.
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20200911-XI-Maths-Sets-2 of 2-Ppt.pdf

The document discusses key concepts about sets, including:
1) Intervals are subsets of real numbers that can be open, closed, or half-open/half-closed. Intervals are represented visually on a number line.
2) The power set of a set A contains all possible subsets of A. Its size is 2 to the power of the size of A.
3) The union of sets A and B contains all elements that are in A, B, or both. The intersection contains all elements common to both sets.
4) Practical problems can use formulas involving the sizes of unions and intersections of finite sets.

Set Concepts

1) Set theory helps organize things into groups and understand logic. Key contributors include Georg Cantor, John Venn, George Boole, and Augustus DeMorgan.
2) A set is a collection of elements. A subset contains only elements that are also in another set. The cardinality of a set refers to the number of elements it contains.
3) Venn diagrams show relationships between sets using overlapping circles to represent their common elements.

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2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(osu毕业证书)美国俄勒冈州立大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。

IEEE Aerospace and Electronic Systems Society as a Graduate Student Member

IEEE Aerospace and Electronic Systems Society as a Graduate Student Member

一比一原版(uofo毕业证书)美国俄勒冈大学毕业证如何办理

原版一模一样【微信：741003700 】【(uofo毕业证书)美国俄勒冈大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(uofo毕业证书)美国俄勒冈大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。

Mechanical Engineering on AAI Summer Training Report-003.pdf

Mechanical Engineering PROJECT REPORT ON SUMMER VOCATIONAL TRAINING
AT MBB AIRPORT

AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...

AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...Paris Salesforce Developer Group

Build the Next Generation of Apps with the Einstein 1 Platform.
Rejoignez Philippe Ozil pour une session de workshops qui vous guidera à travers les détails de la plateforme Einstein 1, l'importance des données pour la création d'applications d'intelligence artificielle et les différents outils et technologies que Salesforce propose pour vous apporter tous les bénéfices de l'IA.Mechatronics material . Mechanical engineering

Mechatronics is a multidisciplinary field that refers to the skill sets needed in the contemporary, advanced automated manufacturing industry. At the intersection of mechanics, electronics, and computing, mechatronics specialists create simpler, smarter systems. Mechatronics is an essential foundation for the expected growth in automation and manufacturing.
Mechatronics deals with robotics, control systems, and electro-mechanical systems.

SCALING OF MOS CIRCUITS m .pptx

this ppt explains about scaling parameters of the mosfet it is basically vlsi subject

NATURAL DEEP EUTECTIC SOLVENTS AS ANTI-FREEZING AGENT

NATURAL DEEP EUTECTIC SOLVENTS AS ANTI-FREEZING AGENT

一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理

原版一模一样【微信：741003700 】【(爱大毕业证书)爱荷华大学毕业证成绩单】【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】外观非常简单，由纸质材料制成，上面印有校徽、校名、毕业生姓名、专业等信息。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
校徽：象征着学校的荣誉和传承。
校名:学校英文全称
授予学位：本部分将注明获得的具体学位名称。
毕业生姓名：这是最重要的信息之一，标志着该证书是由特定人员获得的。
颁发日期：这是毕业正式生效的时间，也代表着毕业生学业的结束。
其他信息：根据不同的专业和学位，可能会有一些特定的信息或章节。
办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】价值很高，需要妥善保管。一般来说，应放置在安全、干燥、防潮的地方，避免长时间暴露在阳光下。如需使用，最好使用复印件而不是原件，以免丢失。
综上所述，办理(爱大毕业证书)爱荷华大学毕业证【微信：741003700 】是证明身份和学历的高价值文件。外观简单庄重，格式统一，包括重要的个人信息和发布日期。对持有人来说，妥善保管是非常重要的。

Software Engineering and Project Management - Introduction, Modeling Concepts...

Introduction, Modeling Concepts and Class Modeling: What is Object orientation? What is OO development? OO Themes; Evidence for usefulness of OO development; OO modeling history. Modeling
as Design technique: Modeling, abstraction, The Three models. Class Modeling: Object and Class Concept, Link and associations concepts, Generalization and Inheritance, A sample class model, Navigation of class models, and UML diagrams
Building the Analysis Models: Requirement Analysis, Analysis Model Approaches, Data modeling Concepts, Object Oriented Analysis, Scenario-Based Modeling, Flow-Oriented Modeling, class Based Modeling, Creating a Behavioral Model.

Pressure Relief valve used in flow line to release the over pressure at our d...

Pressure Relief valve used in flow line to release the over pressure at our desired pressure

Design and optimization of ion propulsion drone

Electric propulsion technology is widely used in many kinds of vehicles in recent years, and aircrafts are no exception. Technically, UAVs are electrically propelled but tend to produce a significant amount of noise and vibrations. Ion propulsion technology for drones is a potential solution to this problem. Ion propulsion technology is proven to be feasible in the earth’s atmosphere. The study presented in this article shows the design of EHD thrusters and power supply for ion propulsion drones along with performance optimization of high-voltage power supply for endurance in earth’s atmosphere.

Applications of artificial Intelligence in Mechanical Engineering.pdf

Historically, mechanical engineering has relied heavily on human expertise and empirical methods to solve complex problems. With the introduction of computer-aided design (CAD) and finite element analysis (FEA), the field took its first steps towards digitization. These tools allowed engineers to simulate and analyze mechanical systems with greater accuracy and efficiency. However, the sheer volume of data generated by modern engineering systems and the increasing complexity of these systems have necessitated more advanced analytical tools, paving the way for AI.
AI offers the capability to process vast amounts of data, identify patterns, and make predictions with a level of speed and accuracy unattainable by traditional methods. This has profound implications for mechanical engineering, enabling more efficient design processes, predictive maintenance strategies, and optimized manufacturing operations. AI-driven tools can learn from historical data, adapt to new information, and continuously improve their performance, making them invaluable in tackling the multifaceted challenges of modern mechanical engineering.

Introduction to verilog basic modeling .ppt

Digital CKT

CEC 352 - SATELLITE COMMUNICATION UNIT 1

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Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...

Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...

Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024

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Software Engineering and Project Management - Software Testing + Agile Method...

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5G Radio Network Througput Problem Analysis HCIA.pdf

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一比一原版(osu毕业证书)美国俄勒冈州立大学毕业证如何办理

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IEEE Aerospace and Electronic Systems Society as a Graduate Student Member

IEEE Aerospace and Electronic Systems Society as a Graduate Student Member

一比一原版(uofo毕业证书)美国俄勒冈大学毕业证如何办理

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Mechanical Engineering on AAI Summer Training Report-003.pdf

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AI + Data Community Tour - Build the Next Generation of Apps with the Einstei...

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Mechatronics material . Mechanical engineering

Mechatronics material . Mechanical engineering

SCALING OF MOS CIRCUITS m .pptx

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UNIT 4 LINEAR INTEGRATED CIRCUITS-DIGITAL ICS

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NATURAL DEEP EUTECTIC SOLVENTS AS ANTI-FREEZING AGENT

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一比一原版(爱大毕业证书)爱荷华大学毕业证如何办理

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Software Engineering and Project Management - Introduction, Modeling Concepts...

Software Engineering and Project Management - Introduction, Modeling Concepts...

Pressure Relief valve used in flow line to release the over pressure at our d...

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Design and optimization of ion propulsion drone

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Applications of artificial Intelligence in Mechanical Engineering.pdf

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Introduction to verilog basic modeling .ppt

- 1. Discrete Mathematics BCSC1010 Module 1 Dr. Praveen Mittal Sets(Lecture2) Lecture Notes of Dr. Praveen Mittal
- 2. Cardinality of a set The cardinality of a set is a measure of a set's size, i.e. the number of elements in the set. If S is a set with n elements, where n is a nonnegative integer, then n is the cardinality of S. It is denoted by |S|. Thus, |S|=n Lecture Notes of Dr. Praveen Mittal
- 3. Cardinality of a set (Examples) The set A = { 1 , 2 , 4 } has a cardinality of 3 Let A be the set of odd positive integers less than 10 Then IAI = 5 Let S be the set of letters in the English alphabets Then lSI = 26 Lecture Notes of Dr. Praveen Mittal
- 4. Cartesian Product Let A and B be sets The Cartesian product of A and B is the set of all ordered pairs (a, b), where a ϵ A and b ϵ B It is denoted by A x B Hence, A x B = {(a, b) | a ϵ A and b ϵ B} Lecture Notes of Dr. Praveen Mittal
- 5. Cartesian Product (Examples) Example − If we take two sets A={a, b} and B={1, 2}, The Cartesian product of A and B is written as A×B={(a,1),(a,2),(b,1),(b,2)} The Cartesian product of B and A is written as B×A={(1,a),(1,b),(2,a),(2,b)} Note: A × B is not the same as B × A. The cardinality of A × B is N*M, where N is the cardinality of A and M is the cardinality of B. Lecture Notes of Dr. Praveen Mittal
- 6. What is the Cartesian product A x B x C where A = {0, 1}, B = {1, 2}, and C = {0, 1, 2}? Solution The Cartesian product A x B x C consists of all ordered triples (a, b, c), where a ϵ A, b ϵ B, and c ϵ C. Hence, A x B x C = {(0, 1,0), (0, 1, 1), (0, 1,2), (0,2,0), (0, 2, 1), (0, 2, 2), (1, 1,0), (1, 1, 1), (1, 1,2), (1, 2, 0), (1, 2, 1), (1, 2, 2)} Lecture Notes of Dr. Praveen Mittal
- 7. Graphical representation of Sets Sets can be represented graphically using Venn diagrams. Named after the English mathematician John Venn, who introduced their use in 1881. In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle. Lecture Notes of Dr. Praveen Mittal
- 8. Graphical representation of Sets Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set. Venn diagrams are often used to indicate the relationships between sets. Lecture Notes of Dr. Praveen Mittal
- 9. Venn diagram Venn diagram that represents V, the set of vowels in the English alphabet. Lecture Notes of Dr. Praveen Mittal
- 10. Set Operations Union Intersection Set Difference Complement Lecture Notes of Dr. Praveen Mittal
- 11. Union of Sets Let A and B be sets The union of the sets A and B is the set that contains those elements that are either in A or in B, or in both It is denoted by A U B AUB = {x | x A or x B} Lecture Notes of Dr. Praveen Mittal
- 12. Union of Sets (Examples) if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} Then A B = {1, 2, 3, 4, 5, 6, 7} The union of the sets {I, 3, 5} and {I, 2, 3} {I, 3, 5} U {I, 2, 3} = {I, 2, 3, 5} Lecture Notes of Dr. Praveen Mittal
- 13. Union of Sets (Graphical Representation) Lecture Notes of Dr. Praveen Mittal
- 14. = {3, 5, 7, 2} Lecture Notes of Dr. Praveen Mittal
- 15. Intersection of Sets Let A and B be sets The intersection of the sets A and B is the set containing those elements in both A and B It is denoted by A B A B = {x | x A and x B} Lecture Notes of Dr. Praveen Mittal
- 16. Intersection of Sets (Example) The intersection of the sets {1, 3, 5} and {1, 2, 3} {1, 3, 5} {l, 2, 3} = {l,3} Lecture Notes of Dr. Praveen Mittal
- 17. Intersection of Sets Lecture Notes of Dr. Praveen Mittal
- 18. = {3, 5} Lecture Notes of Dr. Praveen Mittal
- 19. Difference of Sets Let A and B be two sets The difference of A and B is the set containing those elements that are in A but not in B It is denoted by A - B A - B = {x | x A and x B} Lecture Notes of Dr. Praveen Mittal
- 20. Difference of Sets (Examples) {1, 3, 5} - {1, 2, 3} = {5} { 1, 2, 3} - {1, 3, 5}= {2} Lecture Notes of Dr. Praveen Mittal
- 21. B B Example If A={10,11,12,13} and B={13,14,15}, then (A−B)={10,11,12} and (B−A)={14,15}. Here, we can see (A−B)≠(B−A) Lecture Notes of Dr. Praveen Mittal
- 22. = {7} = {2} Lecture Notes of Dr. Praveen Mittal
- 23. Complement of a set Let U be the universal set. The complement of the set A, denoted by , is the complement of A with respect to U. In other words, the complement of the set A is U - A. An element belongs to if and only if x A. = {x U : x A} Lecture Notes of Dr. Praveen Mittal
- 24. Complement of a set Lecture Notes of Dr. Praveen Mittal
- 25. Complement of a set (Examples) Let A = {a, e, i, o, u} (where the universal set is the set of letters of the English alphabet). Then, = {b, c, d, j, g, h, j, k, I, m, n, p, q, r, s, t, v, w, x, y, z} Let A be the set of positive integers greater than 10 Lecture Notes of Dr. Praveen Mittal
- 26. Disjoint Sets Two sets A and B are called disjoint sets if they do not have even one element in common. Two sets are called disjoint if their intersection is the empty set. Example Let A = {l, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10} Since, A B = Φ, A and B are disjoint Lecture Notes of Dr. Praveen Mittal
- 27. Disjoint Sets Lecture Notes of Dr. Praveen Mittal
- 28. Generalized Unions The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. To denote the union of the sets , ... , , we use the notation Lecture Notes of Dr. Praveen Mittal
- 29. Generalized Intersections The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection. To denote the intersection of the sets , ... , , we use the notation Lecture Notes of Dr. Praveen Mittal
- 30. Lecture Notes of Dr. Praveen Mittal Example
- 31. Next Topic… Proof Techniques Lecture Notes of Dr. Praveen Mittal