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3.1: Set Theory - Mathematics


It is natural for us to classify items into groups, or sets, and consider how those sets overlap with each other. We can use these sets understand relationships between groups, and to analyze survey data.

Basics

An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.

Set

A set is a collection of distinct objects, called elements of the set

A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.

Example 1

Some examples of sets defined by describing the contents:

  1. The set of all even numbers
  2. The set of all books written about travel to Chile

Answers

Some examples of sets defined by listing the elements of the set:

  1. {1, 3, 9, 12}
  2. {red, orange, yellow, green, blue, indigo, purple}

A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.

Notation

Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.

The symbol ∈ means “is an element of”.

A set that contains no elements, { }, is called the empty set and is notated ∅

Example 2

Let A = {1, 2, 3, 4}

To notate that 2 is element of the set, we’d write 2 ∈ A

Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a subset of the larger set of all Madonna albums.

Subset

A subset of a set A is another set that contains only elements from the set A, but may not contain all the elements of A.

If B is a subset of A, we write BA

A proper subset is a subset that is not identical to the original set—it contains fewer elements.

If B is a proper subset of A, we write BA

Example 3

Consider these three sets:

A = the set of all even numbers
B = {2, 4, 6}
C = {2, 3, 4, 6}

Here BA since every element of B is also an even number, so is an element of A.

More formally, we could say BA since if x B, then x A.

It is also true that BC.

C is not a subset of A, since C contains an element, 3, that is not contained in A

Example 4

Suppose a set contains the plays “Much Ado About Nothing,” “MacBeth,” and “A Midsummer’s Night Dream.” What is a larger set this might be a subset of?

There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.

Try It Now

The set A = {1, 3, 5}. What is a larger set this might be a subset of?

Union, Intersection, and Complement

Commonly sets interact. For example, you and a new roommate decide to have a house party, and you both invite your circle of friends. At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets.

Union, Intersection, and Complement

The union of two sets contains all the elements contained in either set (or both sets). The union is notated A B. More formally, x A B if x A or x B (or both)

The intersection of two sets contains only the elements that are in both sets. The intersection is notated A B. More formally, x A B if x A and x B.

The complement of a set A contains everything that is not in the set A. The complement is notated A’, or Ac, or sometimes ~A.

Example 5

Consider the sets:

A = {red, green, blue}
B = {red, yellow, orange}
C = {red, orange, yellow, green, blue, purple}

Find the following:

  1. Find A B
  2. Find A B
  3. Find AcC

Answers

  1. The union contains all the elements in either set: A B = {red, green, blue, yellow, orange} Notice we only list red once.
  2. The intersection contains all the elements in both sets: A B = {red}
  3. Here we’re looking for all the elements that are not in set A and are also in C. AcC = {orange, yellow, purple}

Try It Now

Using the sets from the previous example, find A C and BcA

Notice that in the example above, it would be hard to just ask for Ac, since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set. For this reason, complements are usually only used with intersections, or when we have a universal set in place.

Universal Set

A universal set is a set that contains all the elements we are interested in. This would have to be defined by the context.

A complement is relative to the universal set, so Ac contains all the elements in the universal set that are not in A.

Example 6

  1. If we were discussing searching for books, the universal set might be all the books in the library.
  2. If we were grouping your Facebook friends, the universal set would be all your Facebook friends.
  3. If you were working with sets of numbers, the universal set might be all whole numbers, all integers, or all real numbers

Example 7

Suppose the universal set is U = all whole numbers from 1 to 9. If A = {1, 2, 4}, then Ac= {3, 5, 6, 7, 8, 9}.

As we saw earlier with the expression AcC, set operations can be grouped together. Grouping symbols can be used like they are with arithmetic – to force an order of operations.

Example 8

Suppose H = {cat, dog, rabbit, mouse}, F = {dog, cow, duck, pig, rabbit}, and W = {duck, rabbit, deer, frog, mouse}

  1. Find (H F) ⋃ W
  2. Find H ⋂ (FW)
  3. Find (H F)cW

Solutions

  1. We start with the intersection: H F = {dog, rabbit}. Now we union that result with W: (H F) ⋃ W = {dog, duck, rabbit, deer, frog, mouse}
  2. We start with the union: FW = {dog, cow, rabbit, duck, pig, deer, frog, mouse}. Now we intersect that result with H: H ⋂ (FW) = {dog, rabbit, mouse}
  3. We start with the intersection: H F = {dog, rabbit}. Now we want to find the elements of W that are not in H F. (H F)cW = {duck, deer, frog, mouse}

Venn Diagrams

To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the eighteenth century. These illustrations now called Venn Diagrams.

Venn Diagram

A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets.

Basic Venn diagrams can illustrate the interaction of two or three sets.

Example 9

Create Venn diagrams to illustrate A B, A B, and AcB

A B contains all elements in either set.

A B contains only those elements in both sets—in the overlap of the circles.

Acwill contain all elements not in the set A. AcB will contain the elements in set B that are not in set A.

Example 10

Use a Venn diagram to illustrate (H F)cW

We’ll start by identifying everything in the set H F

Now, (H F)cW will contain everything not in the set identified above that is also in set W.

Example 11

Create an expression to represent the outlined part of the Venn diagram shown.

The elements in the outlined set are in sets H and F, but are not in set W. So we could represent this set as H FWc

Try It Now

Create an expression to represent the outlined portion of the Venn diagram shown

Cardinality

Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set.

Cardinality

The number of elements in a set is the cardinality of that set.

The cardinality of the set A is often notated as |A| or n(A)

Example 12

Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}.

What is the cardinality of B? AB, A B?

Answers

The cardinality of B is 4, since there are 4 elements in the set.

The cardinality of AB is 7, since AB = {1, 2, 3, 4, 5, 6, 8}, which contains 7 elements.

The cardinality of A B is 3, since A B = {2, 4, 6}, which contains 3 elements.

Example 13

What is the cardinality of P = the set of English names for the months of the year?

Answers

The cardinality of this set is 12, since there are 12 months in the year.

Sometimes we may be interested in the cardinality of the union or intersection of sets, but not know the actual elements of each set. This is common in surveying.

Example 14

A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:

  • Tea only
  • Coffee only
  • Both coffee and tea

Suppose 20 report tea only, 80 report coffee only, 40 report both. How many people drink tea in the morning? How many people drink neither tea or coffee?

Answers

This question can most easily be answered by creating a Venn diagram. We can see that we can find the people who drink tea by adding those who drink only tea to those who drink both: 60 people.

We can also see that those who drink neither are those not contained in the any of the three other groupings, so we can count those by subtracting from the cardinality of the universal set, 200.

200 – 20 – 80 – 40 = 60 people who drink neither.

Example 15

A survey asks: “Which online services have you used in the last month?”

  • Twitter
  • Facebook
  • Have used both

The results show 40% of those surveyed have used Twitter, 70% have used Facebook, and 20% have used both. How many people have used neither Twitter or Facebook?

Answers

Let T be the set of all people who have used Twitter, and F be the set of all people who have used Facebook. Notice that while the cardinality of F is 70% and the cardinality of T is 40%, the cardinality of FT is not simply 70% + 40%, since that would count those who use both services twice. To find the cardinality of FT, we can add the cardinality of F and the cardinality of T, then subtract those in intersection that we’ve counted twice. In symbols,

n(FT) = n(F) + n(T) – n(FT)
n(FT) = 70% + 40% – 20% = 90%

Now, to find how many people have not used either service, we’re looking for the cardinality of (FT)c . Since the universal set contains 100% of people and the cardinality of FT = 90%, the cardinality of (FT)c must be the other 10%.

The previous example illustrated two important properties

Cardinality properties

n(AB) = n(A) + n(B) – n(AB)

n(Ac) = n(U) – n(A)

Notice that the first property can also be written in an equivalent form by solving for the cardinality of the intersection:

n(AB) = n(A) + n(B) – n(AB)

Example 16

Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the next quarter.

21 were taking a SS course26 were taking a HM course
19 were taking a NS course9 were taking SS and HM
7 were taking SS and NS10 were taking HM and NS
3 were taking all three7 were taking none

How many students are only taking a SS course?

Answers

It might help to look at a Venn diagram. From the given data, we know that there are 3 students in region e and 7 students in region h.

Since 7 students were taking a SS and NS course, we know that n(d) + n(e) = 7. Since we know there are 3 students in region 3, there must be 7 – 3 = 4 students in region d.

Similarly, since there are 10 students taking HM and NS, which includes regions e and f, there must be 10 – 3 = 7 students in region f.

Since 9 students were taking SS and HM, there must be 9 – 3 = 6 students in region b.

Now, we know that 21 students were taking a SS course. This includes students from regions a, b, d, and e. Since we know the number of students in all but region a, we can determine that 21 – 6 – 4 – 3 = 8 students are in region a.

8 students are taking only a SS course.

Try It Now

One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.

43 believed in UFOs44 believed in ghosts
25 believed in Bigfoot10 believed in UFOs and ghosts
8 believed in ghosts and Bigfoot5 believed in UFOs and Bigfoot
2 believed in all three

How many people surveyed believed in at least one of these things?


A set is an unordered collection of different elements. A set can be written explicitly by listing its elements using set bracket. If the order of the elements is changed or any element of a set is repeated, it does not make any changes in the set.

Some Example of Sets

  • A set of all positive integers
  • A set of all the planets in the solar system
  • A set of all the states in India
  • A set of all the lowercase letters of the alphabet

Principles of Mathematics: A Primer

Books that introduce undergraduate students to higher mathematics are numerous. One common complaint against a significant portion of them is that they talk a lot about how to prove theorems without actually proving anything interesting. It is as if a dinner host discussed the details of eating a nice meal, but only served chips and water.

Some authors avoid that trap by choosing one part of higher mathematics and using that part to show the students the power of their new theorem-proving tools. The book under review goes down this path, choosing group theory and linear algebra to be the areas in which substantial results will be proved in Chapters 5 and 6 (the last two chapters in the book). The author goes deep enough into group theory to cover normal subgroups and isomorphism theorems, and deep enough into linear algebra to discuss eigenvalue theory and the general linear group.

A few choices that the author makes in ordering his early chapters are rather surprising. Chapter 3, which starts on page 183, is called Proofs, but by then we have done many proofs which were called just that. Functions are the topic of Chapter 4, but Set Theory is Chapter 1, and in that first chapter, we prove, for instance, that the rationals are equinumerous to the integers, using bijective functions of course. We also prove that there is no bijection between the reals and the integers.

These choices can perhaps be explained in a somewhat contrived way, for instance by saying that students already have an idea of what a function is, what proofs are, and we will formalize these concepts later, but this reviewer thinks that a book that intends to introduce readers into the world of proof-based mathematics is not the place for this kind of non-linear coverage.

There are plenty of exercises and supplementary problems, though none of them come with solutions. Many sentences that end in a math formula do not have a period at the end, which sometimes makes reading harder than it should be. To summarize, the book is certainly different from the competition, but more editing, and especially a more straightforward ordering of the topics, would have improved it.

Miklós Bóna is Professor of Mathematics at the University of Florida.

1.2 Set Theory &ndash Definitions, Notation, and Terminology &ndash What is a Set?, 3


Set Theory & Algebra

A binary operation on a set of integers is defined as x y = x 2 + y 2 . Which one of the following statements is TRUE about ? Associativity: A binary operation ∗ on a set S is said to be associative if it satisfies the associative law: a ∗ (b ∗c) = (a ∗b) ∗c for all a, b, c ∈S. Commutativity: A binary operation ∗ on a set S is said to be commutative if it satisfies the condition: a ∗b=b ∗a for all a, b, ∈S. In this case, the order in which elements are combined does not matter. Solution: Here a binary operation on a set of integers is defined as x⊕ y = x2 + y2. for Commutativity: x ⊕y= y ⊕x. LHS=> x ⊕y= x^2+ y^2 RHS=> y ⊕x= y^2+x^2 LHS = RHS. hence commutative. for Associativity: x ⊕ (y ⊕ z) =(x ⊕ y) ⊕ z LHS=> x ⊕ (y⊕ z) = x ⊕ ( y^2+z^2)= x^2+(y^2+z^2)^2 RHS=> (x ⊕y) ⊕z= ( x^2+y^2) ⊕z=(x^2+y^2)^2+z^2 So, LHS ≠ RHS, hence not associative. Reference: http://faculty.atu.edu/mfinan/4033/absalg3.pdf This solution is contributed by Nitika Bansal Another Solution : commutative as xy is always same as yx. is not associative as (xy)z is (x^2 + y^2)^2 + z^2, but x(yz) is x^2 + (y^2 + z^2)^2.

Introduction to naive set theory

In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. To indicate that an object x is a member of a set A one writes xA, while xA indicates that x is not a member of A. A set may be defined by a membership rule (formula) or by listing its members within braces. For example, the set given by the rule “prime numbers less than 10” can also be given by <2, 3, 5, 7>. In principle, any finite set can be defined by an explicit list of its members, but specifying infinite sets requires a rule or pattern to indicate membership for example, the ellipsis in <0, 1, 2, 3, 4, 5, 6, 7, …>indicates that the list of natural numbers ℕ goes on forever. The empty (or void, or null) set, symbolized by <> or Ø, contains no elements at all. Nonetheless, it has the status of being a set.

A set A is called a subset of a set B (symbolized by AB) if all the members of A are also members of B. For example, any set is a subset of itself, and Ø is a subset of any set. If both AB and BA, then A and B have exactly the same members. Part of the set concept is that in this case A = B that is, A and B are the same set.


Fall 2012

Please be aware that practice exams are longer in length than the actual exams. They are meant to give an idea of the material to be covered.

  • Chapter 2 - Set Theory
      (8/30/12)
  • 2.3 - Set Operations (9/4/12 & 9/6/12) (9/11/12 & 9/13/12)
  • Review (9/18/12)
    • 3.1 - Statements, Connectives, & Quantifiers (9/25/12)
    • 3.2 - Truth Tables (9/27/12)
    • 3.3 - The Conditional & Biconditional (10/2/12)
    • 3.4 - Verifying Arguments (10/4/12)
    • 3.5 - Using Euler Diagrams to Verify Syllogisms (10/9/12)
    • Review (10/11/12)
    • 13.1 - Basic Counting Methods (10/23/12)
    • 13.2 - Fundamental Counting Principle (10/25/12)
    • 13.3 - Permutations and Combinations (10/30/12)
    • 14.1 - Basic Probability Theory (11/1/12)
    • 14.2 - Complements and Unions of Events (11/6/12)
    • 14.3 - Conditional Probability and Intersections of Events (11/8/12)
    • 14.4 - Expected Value (11/13/12)
    • Review (11/15/12)
    • 15.1 - Organizing and Visualizing Data (11/27/12)
    • 15.2 - Measures of Central Tendency (11/29/12)
    • 15.3 - Measures of Dispersion (12/4/12)
    • 15.4 - The Normal Distribution (12/6/12)
    • Review (12/11/12)

    This page maintained by Jess Lenarz (jessie "dot" lenarz "at" mnstate "dot" edu)


    3.1: Set Theory - Mathematics

    Nelson Principles of Mathematics
    Tables of Contents

    Nelson Principles of Mathematics 11

    Chapter 1 – Inductive and Deductive Reasoning
    Getting Started: The Mystery of the Mary Celeste
    1.1 Making Conjectures: Inductive Reasoning
    1.2 Exploring the Validity of Conjectures
    1.3 Using Reasoning to Find a Counterexample to a Conjecture

    Applying Problem-Solving Strategies: Analyzing a Number Puzzle

    1.4 Proving Conjectures: Deductive Reasoning
    Mid-Chapter Review
    1.5 Proofs That Are Not Valid
    1.6 Reasoning to Solve Problems
    1.7 Analyzing Puzzles and Games
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 1: Creating an Action Plan

    Chapter 2 – Properties of Angles and Triangles
    Getting Started: Geometric Art
    2.1 Exploring Parallel Lines
    2.2 Angles Formed by Parallel Lines

    Applying Problem-Solving Strategies: Checkerboard Quadrilaterals

    Mid-Chapter Review
    2.3 Angle Properties in Triangles
    2.4 Angle Properties in Polygons

    2.5 Exploring Congruent Triangles

    2.6 Proving Congruent Triangles
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 2: Selecting Your Research Topic
    Cumulative Review 1

    Chapter 3 – Acute Triangle Trigonometry
    Getting Started: Lacrosse Trigonometry
    3.1 Exploring Side–Angle Relationships in Acute Triangles
    3.2 Proving and Applying the Sine Law
    Mid-Chapter Review
    3.3 Proving and Applying The Cosine Law
    3.4 Solving Problems Using Acute Triangles
    Applying Problem-Solving Strategies: Analyzing a Trigonometry Puzzle
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 3: Creating Your Research Question or Statement

    Chapter 4 – Radicals
    Getting Started: Photography
    4.1 Mixed and Entire Radicals
    4.2 Adding and Subtracting Radicals

    4.3 Multiplying and Dividing Radicals

    Applying Problem-Solving Strategies: Defining a Fractal
    Mid-Chapter Review
    4.4 Simplifying Algebraic Expressions Involving Radicals

    4.5 Exporing Radical Equations

    4.6 Solving Radical Equations
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 4: Carrying OutYour Research

    Chapter 5 – Statistical Reasoning
    Getting Started: Comparing Salaries
    5.1 Exploring Data
    5.2 Frequency Tables, Histograms, and Frequency Polygons

    5.4 The Normal Distribution
    Applying Problem-Solving Strategies: Predicting Possible Pathways
    5.5 Z-Scores
    5.6 Confidence Intervals
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 5: Analyzing Your Data
    Cumulative Review 2

    Chapter 6 – Quadratic Functions
    Getting Started: String Art

    6.1 Exploring Quadratic Relations
    6.2 Properties of Graphs of Quadratic Functions
    6.3 Factored Form of a Quadratic Function
    Mid-Chapter Review
    6.4 Vertex Form of a Quadratic Function
    6.5 Solving Problems Using Quadratic Function Models
    Applying Problem-Solving Strategies: Curious Counting Puzzles

    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 6: Identifying Controversial Issues

    Chapter 7 – Quadratic Equations
    Getting Started: Factoring Design
    7.1 Solving Quadratic Equations by Graphing
    7.2 Solving Quadratic Equations by Factoring
    7.3 Solving Quadratic Equations by Using the Quadratic Formula
    Mid-Chapter Review
    7.4 Solving Problems Using Quadratic Equations
    Applying Problem-Solving Strategies: Determining Quadratic Patterns
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 7: The Final Product and Presentation

    Chapter 8 - Proportional Reasoning
    Getting Started: Interpreting the Cold Lake Region
    8.1 Comparing and Interpreting Rates
    8.2 Solving Problems That Involve Rates
    Applying Problem Solving Strategies: Analyzing a Rate Puzzle

    8.3 Scale Diagrams
    8.4 Scale Factors and Areas of 2-D Shapes
    8.5 Similar Objects: Scale Models and Scale Diagrams
    8.6 Scale Factors and 3-D Objects
    Chapter Self-Test
    Chapter Review
    Chapter Task
    Project Connection 8: Peer Critiquing of Research Projects
    Cumulative Review 3

    Nelson Principles of Mathematics 12

    Chapter 1: Set Theory 2
    Getting Started: Real Estate Listings 4
    1.1 Types of Sets and Set Notation 6
    1.2 Exploring Relationships between Sets 19
    1.3 Intersection and Union of Two Sets 22
    History Connection: Unexpected Infinities 35
    Mid-Chapter Review 36
    1.4 Applications of Set Theory 39
    Math in Action: Relevant Hits 54
    Applying Problem-Solving Strategies: Analyzing a Logic Puzzle 55
    Chapter Self-Test 56
    Chapter Review 57
    Chapter Task: Planning a Zoo 59
    Project Connection: Creating an Action Plan 60

    Chapter 2: Counting Methods 62
    Getting Started: The Tower of Hanoi 64
    2.1 Counting Principles 66
    2.2 Introducing Permutations and Factorial Notation 76
    2.3 Permutations When All Objects are Distinguishable 84
    Math in Action: Birthday Permutations 95
    Mid-Chapter Review 96
    2.4 Permutations When Objects Are Identical 98
    Applying Problem-Solving Strategies: Disk Drop 108
    2.5 Exploring Combinations 109
    2.6 Combinations 111
    2.7 Solving Counting Problems 121
    History Connection: Computer Codes 128
    Chapter Self-Test 129
    Chapter Review 130
    Chapter Task: Analyzing a Traditional Game 133
    Project Connection: Selecting Your Research Topic 134

    Chapter 3: Probability 136
    Getting Started: Dice Differences 138
    3.1 Exploring Probability 140
    3.2 Probability and Odds 142
    3.3 Probabilities Using Counting Methods 151
    History Connection: Counter Intuition 162
    Mid-Chapter Review 163
    3.4 Mutually Exclusive Events 166
    Applying Problem-Solving Strategies: The Monty Hall Puzzle 181
    3.5 Conditional Probability 182
    3.6 Independent Events 192
    Math in Action: Modelling with Probabilities 201
    Chapter Self-Test 202
    Chapter Review 203
    Chapter Task: Games and Probability 207
    Project Connection: Creating Your Research Question or Statement 208
    Chapters 1-3 Cumulative Review 210

    Chapter 4: Rational Expressions and Equations 212
    Getting Started: Comparing Internet Plans 214
    4.1 Equivalent Rational Expressions 216
    4.2 Simplifying Rational Expressions 225
    4.3 Multiplying and Dividing Rational Expressions 232
    Mid-Chapter Review 240
    4.4 Adding and Subtracting Rational Expressions 244
    Applying Problem-Solving Strategies: Exploring Rational Expressions 251
    History Connection: The Thin Lens Formula 252
    4.5 Solving Rational Equations 253
    Math in Action: The Harmonic Mean 261
    Chapter Self-Test 262
    Chapter Review 263
    Chapter Task: The Rational Expressions Dice Game 267
    Project Connection: Carrying Out Your Research 268

    Chapter 5: Polynomial Functions 270
    Getting Started: Staging a Profitable Production 272
    5.1 Exploring the Graphs of Polynomial Functions 274
    5.2 Characteristics of the Equations of Polynomial Functions 278
    Math in Action: Motion Due to Gravity 289
    Applying Problem-Solving Strategies: The Poly-Nomial Game 292
    Mid-Chapter Review 293
    5.3 Modelling Data with a Line of Best Fit 295
    5.4 Modelling Data with a Curve of Best Fit 307
    History Connection: Prime-Generating Quadratic Polynomials 317
    Chapter Self-Test 318
    Chapter Review 319
    Chapter Task: Experimenting with Polynomial Models 323
    Project Connection: Analyzing Your Data 324
    Chapters 4-5 Cumulative Review 327

    Chapter 6: Exponential Functions 330
    Getting Started: Origami 332
    6.1 Exploring the Characteristics of Exponential Functions 334
    6.2 Relating the Characteristics of an Exponential Function to Its Equation 338
    6.3 Solving Exponential Equations 352
    Mid-Chapter Review 366
    6.4 Modelling Data Using Exponential Functions 370
    History Connection: Ernest Rutherford (1871-1937) 384
    Applying Problem-Solving Strategies: Exponential Hit 385
    6.5 Financial Applications Involving Exponential Functions 386
    Math in Action: The Rule of 72 399
    Chapter Self-Test 400
    Chapter Review 401
    Chapter Task: Caffeine Decay 405
    Project Connection: Identifying Controversial Issues 406

    Chapter 7: Logarithmic Functions 408
    Getting Started: Fibre Optic Cables 410
    7.1 Characteristics of Logarithmic Functions with Base 10 and Base e 412
    7.2 Evaluating Logarithmic Expressions 426
    Mid-Chapter Review 439
    7.3 Laws of Logarithms 442
    Math in Action: Making a Slide Rule 448
    7.4 Solving Exponential Equations Using Logarithms 449
    History Connection: Euler’s Number 459
    7.5 Modelling Data Using Logarithmic Functions 460
    Applying Problem-Solving Strategies: Guess the Number: Binary Search 472
    Chapter Self-Test 473
    Chapter Review 474
    Chapter Task: First Digits 477
    Project Connection: The Final Product and Presentation 478

    Chapter 8: Sinusoidal Functions 480
    Getting Started: Sine and Cosine Patterns 482
    8.1 Understanding Angles 484
    8.2 Exploring Graphs of Periodic Functions 491
    History Connection: Not as Easy as ! 496
    8.3 The Graphs of Sinusoidal Functions 497
    Mid-Chapter Review 513
    8.4 The Equations of Sinusoidal Functions 516
    Math in Action: Biorhythms 532
    8.5 Modelling Data with Sinusoidal Functions 533
    Applying Problem-Solving Strategies: Hidden Waves 548
    Chapter Self-Test 549
    Chapter Review 550
    Chapter Task: Here Comes the Sun 553
    Project Connection: Peer Critiquing of Research Projects 554
    Chapters 6-8 Cumulative Review 556


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    Math Assignment Class XI Ch -1 Set Theory

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    Mathematics Assignment

    Class - XI / Subject Mathematics / Chapter - 1

    For Non-Medical /Applied Math Students

    Q1 Write the following sets in tabular form

    Q 2 Write the following in set builder form

    Q3 Write the subsets of the set A = <1, 2, <3>>

    Q4 Write the proper subsets of A =

    Ans: Except < 5, 10, 11, 15>, all other subsets of A are proper subsets.

    Q5 Write the power set of A =

    Q6 List all the subsets of set

    Q7 From the sets given below select the equal and equivalent sets

    Ans Equal sets: A and C, Equivalent Sets : B, E and F A and C

    then find n(S) + n(P) Ans 41

    Q 9 Let A and B are two sets , show A - B , A ∩ B, B - A in venn diagram.

    Q 10 If A = , B = , C = verify that

    (b) A - (B ∩ C) = (A - B) (A - C)

    Q11 If A , B and C are two sets then, prove that

    a) A (A ∩ B) = A

    b) (A B) – B = A - B

    c) (A - B) (A ∩ B = A

    d) A (B - A) = A B

    f) A B) – B = A - B

    Q 12 If A = <1, 2, 3, 4, 5>, B = <1, 3, 5, 7, 9>, C = <2, 3, 4>, verify that

    A - (B C) = (A - B) ∩ (A - C)

    Q 13 In a class of 35 students, 24 like to play cricket and 16 like to play football. Also each student like to play at least one of the two games. How many like to play both Cricket and Football. Ans 5

    Q 14 In a survey of 400 students in the school, 100 were taking apple juice, 150 were taking orange juice. Find how many students were taking neither apple juice not orange juice. Ans 225

    Q 15 A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals went to a total of 58 men and only three men got medals in all the three sports, how many received medals in exactly two of the three sports ? Ans 9

    Q 16 In a group of 200 students, it was found that 120 study maths, 90 study Physics, 70 study Chemistry, 40 study Maths and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Maths, 20 study all the three subject. Find the number of students.

    (i) Who study all the three subjects ? Ans 20

    (ii) Who study maths only ? Ans a = 50

    (iii) Who study one of the three subject ? Ans 100

    (iv) Who study two of the three subjects ? Ans 60

    Q 17 Out of 100 students, 15 passed in english, 12 passed in Maths, 8 passed in science, 6 in English and Maths, 7 in Maths and Science, 4 in English and science, 4 in all three. Find how many passed in

    (i) English and Maths but not in science Ans 2

    (ii) Maths and Science but not in English Ans 3

    (iv) More than one subject only Ans 9

    Q 18 In a group of 50 students , 17 students studying French , 13 English, 15 Sanskrit, 9 studying French and English, 4 Studying English and Sanskrit, 5 studying French and Sanskrit, 3 studying all the three subjects. Find the number of students who study.


    Theory of Sets and Its Business Applications

    Set theory plays a vital role in modern mathematics and it is used in other disciplines also.

    3.1 SET THEORY

    A set S is a collection of definite and well-defined objects. These objects are called elements of the set.

    3.2 REPRESENTATION OF SETS

    The representation of a set can be done in two ways:

    3.2.1 Tabulation Method

    The individual elements of a set are listed and separated by using commas and enclosed within brackets.

    A set of natural numbers less than seven is represented as A =

    3.2.2 Set Builder Form

    The set is described by stating a common property of its member.

    The set of odd numbers less than .

    Get Business Mathematics now with O’Reilly online learning.

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    Common Questions and Shit I May Have Missed

    Questions? Yes, you, with the piss-wet pants.

    “But Aaron. I don't have any access to the specialized kit. What am I to do?”

    Sigh. Alright. I get that this is a common problem. Let me say if you're looking to do a competition, find a gym you can travel to at least once every couple weeks during your prep time that at least has some of the equipment you need.

    Dumbbell farmers walks are fine but it’s not like the torpedo they make you run around with on contest day. There is a crossover, but why leave it to chance? A lot can be done with a set of Fat Gripz, your own stretch bands to tie weight plates together to emulate stones or shields, and a beer keg (filled with water or sand) or sandbag in your garden shed. Invest a little. It’s worth it. Skip the next work curry night if money is tight. Fuck those guys, they don’t help you lift cool shit.

    “I can only afford one pair of shoes. What’s the best all round strongman shoe?”

    “I’ve seen a lot of strongmen strapped up like they’re the Stay Puff Marshmallow Man on roids. Do I need straps, sleeves, wraps, belts, etc.?”

    Short answer, yes. But I competed my first 18 months without any assistive gear except straps for deadlift for reps, and I managed to win Britain’s Strongest Novice athlete at BodyPower UK. So you don't need it all in the beginning. However, as the weights get bigger and bigger, it definitely becomes useful and safer to kit up — not for every single set and rep, but on your working sets it definitely doesn’t hurt.

    Hahaha. I like pizza. Look, eat for your goal like you train for your goal. Good meats, tasty carbs, and Budweiser on a Friday.


    Is $ in , , 1, 2>$?

    I recently picked up Robert R. Stoll's book, 'Set Theory and Logic' and whilst reading chapter 1.3, I came into the question posted above. It's salient to note that my mathematical knowledge in general rivals that of a 2nd grader (no offense to any 2nd graders), so this question brought me some trouble.

    From what I learned in the book hitherto, a set is a collection of unique elements or members, so for example: $a in < a, b, c >$ is true because $a$ is an element within the given set. However, the brackets in the original question are throwing me off a bit because I am not too sure whether the $1$ and $2$ have to be within its own brackets e.g. $< 1, 2>.$

    In any case, I would have said that $< 1, 2 >$ is indeed within $< < 1, 2, 3 >, < 1, 3 >, 1, 2 >,$ but that answer is said with soft conviction and some confusion. Hopefully someone with more mathematical prowess can help me. Thanks in advance.


    Watch the video: FM 30 Sets and Set Notation (December 2021).