MMWModule- Chapter 2 Mathematical Language and Symbol PDF

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FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

MODULE 2 MATHEMATICAL LANGUAGE AND SYMBOL MODULE OVERVIEW

This module consist of four lessons : Mathematics and English as Languages , The Language and Grammar of Mathematics, The Language of Sets, The Language of Logic . Each lesson was designed as a self-teaching guide. Definitions of terms and examples had been incorporated. Answering the problems in “your turn” will check your progress. You may compare your answers to the solutions provided at the later part of this module in that way you will be able to measure your achievement and as well as the effectiveness of the module. Exercises were prepared as your assignment to measure your understanding about the topics. MODULE LEARNING OBJECTIVES

At the end of the module, you should be able to:  Discuss the language, symbols and conventions of mathematics  Explain the nature of mathematics as a language  Perform operations on mathematical expressions correctly  Acknowledge that mathematics is a useful language LEARNING CONTENTS ( MATHEMATICS AND ENGLISH LANGUAGES)

Lesson 1. Mathematics and English as Languages Mathematics and English are both languages that if you excel to both, you have a bigger chance of being ahead in a world full of competition. The figure below will give you an overview on their similarities and differences.

Even though the words “noun”, “verb”, or pronoun” are not used in mathematics, the similarities with the English language can be observed :  Nouns could be constants such as numbers or expression with numbers:

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

( 31 ) ,−58

12 , 2 4− 

A verb could be equal sign ¿ , or inequality symbols like



Pronouns could be a variables like x∨ y : −4 5 x−8 , 2 xy , x Sentences could be formed by putting together these parts : , 3 x+7=24 2 x +3 y=7



¿ or ≤ .

LEARNING CONTENTS (THE LANGUAGE AND GRAMMAR OF MATHEMATICS)

Lesson 2: The Language and Grammar of Mathematics Mathematics has a language features unparalleled in other languages, like presentation for example “ , y ,∨ x “ for any real number or any numerical expression. In addition, the language of Mathematics is packed with terms and symbols, which normally used in everyday conversation. Mathematics has grammar too, but it is the mathematical logic that determines whether the statements are true or not true, valid or not valid. Therefore, we need to view Mathematics as a language and must learn it in a way a language is learned.

Think about this! Watch this video and answer the guide questions?

Math isn't hard, it's a language | Randy Palisoc https://www.youtube.com/watch?v=V6yixyiJcos

1. What practice in learning Mathematics was emphasized by the speaker? 2. What part of the talk made you go back to the time when you were beginning to learn Mathematics? 3. Do you agree that Mathematics is a human language , and should have been taught the way English languages is being taught? Explain.

Mathematical Expression and Sentence A sentence must contain a complete thought. In the English language an ordinary sentence must contain a subject and a predicate.. Similarly, a mathematical sentence must state a complete thought. While an expression is a name given to a mathematical object of interest. Below are examples of mathematical expressions : PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

a. An ordered pair 1 4 b. A matrix −2 3 c. A function f (x) d. The set {1, 3, 5} e. Below are example of mathematical sentences or statement.

[

]

Combined Mathematical Sentences 6 +2 =2 4 2( 12 −4 )=16

English Translations - The sum of six and two all over four is two. - The ratio of six plus two, and four is equal to two. - Two is the quotient when the sum of six and two is divided by four. -Twice the difference of twelve and four is sixteen. -The product of 2 and twelve less four is sixteen

We also have some examples of algebraic sentences. Algebraic Sentences 2 x =14

English Translation -Twice a number is equal to fourteen. -Two times a number is fourteen.

3 ( 2 x−1) =4

-Thrice the difference of twice a number and one is four. -Three, multiplied to the difference of twice a number less one ,is four

2( x=1) =5 3

-The ratio of twice the sum of a number and one , and three is equal to five.

Example

Translate the following expression and sentences in English into Mathematical expressions and statement or vice versa. Use the letter n to represent the unknown. a. A number increased by 10 . b. 2n + 1. c. The difference between the ages of mom and son is 27. d. 3 n−1=23 Solution a. n+10 or 10+n b. The sum of twice a number and 1. c. m−n=27 d. The difference of thrice a number and 1 is twenty-three. PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

Your turn

Translate the following expression and sentences in English into Mathematical expressions and statement or vice versa. Use the letter to represent the unknown. a. Four times the square of a number b. (x , y )∈ A c. x 2+1 ≠ 0 d. A sum of three consecutive numbers is eighteen.

x

LEARNING POINTS A sentence must contain a complete thought. In the English language an ordinary sentence must contain a subject and a predicate.. Similarly, a mathematical sentence must state a complete thought. While an expression is a name given to a mathematical object of interest. LEARNING ACTIVITY 1

A. Translate each phrase or sentence into a mathematical expression or equation. Twelve more than a number. _______________________ Eight minus a number. _______________________ An unknown quantity less fourteen. _______________________ Six times a number is fifty-four. _______________________ Two ninths of a number is eleven. _______________________ Three more than seven times a number is nine more than five times the number. _________________ 7. Twice a number less eight is equal to one more than three times the number. _______________________ 8. Six is subtracted from the sum of x and two times y . __________________________ 9. Five times x reduced by the square of y . _________________________________ 10. Subtract the product of x and y from fifty-eight._____________________ 1. 2. 3. 4. 5. 6.

B. Give your own expression and sentences that conform with the stated type and truth value . Take note : None means no verb or connective being used. Open means , need verification / conditions Ee −expression ∈ english E M – expression in mathematics S e −sentence ∈ english S M - sentence in mathematics

Sentences/Expression Ex. 1. 2.

3 + 3 >6

Type SM SE SE

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Verb/Connectives + and >

Truth Value of Sentence: True/False/Open False True False 4

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

EE SM SM EM SM EM SE SM

3. 4. 5. 6. 7. 8. 9. 10.

None True False None None Open Open True

LEARNING CONTENTS ( THE LANGUAGES OF SETS)

Lesson 3. The Language of Sets The concept of sets was formalized by George Cantor, a German mathematician (1845-1918). He defined set as a collection of definite distinguished objects called elements.

3.1 Set Notations It is important to note that in Mathematics , there are certain conventions in the ways sets are represented, written, and interpreted .The following examples will illustrate these conventions. The set is composed of five vowels of the English alphabet may be named and can be denoted as A= { a , e , i, o , u } . Sets like A which has a definite number of elements in roster method where the elements are listed . Commas are used between each element and a pair of braces are used to enclose the list elements.

Example 1

The set whose elements are all integers may be named and can be written as N={ x∨x ∈ Z } . This is read as “Set N is the set of all values of x is an integer”. Sets like Set N which has indefinite number of elements are

Example 2

x such that written in set-builder method . This notation is used whenever it is convenient or impossible to list all the elements of a set; it merely describes the characterizing property of its elements in terms of symbols.

We also have to take note of the frequently used set of numbers :

Your turn

Use The Roster Method to Represent a Set

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

a. The set of natural numbers less than 5. b. The solution set of x+5=−1 c. The set of negative integers greater than

Your turn

Module 2 : Mathematical Language and Symbol

−4

Use The Set-builder Method to Represent a Set

the set whose elements are numbers greater than negative five but less than a. Set M is four . b. Set P is the set whose elements are numbers greater than or equal to zero. c. Set N is the set of even numbers greater than or equal to four but less than or equal to twenty.

More Concepts to Learn about Sets 

Finite Set is a set whose elements are countable. Examples are : o A={¿ , orange, ¿ , ¿ , ¿ } . o Set B is the set of integers greater than zero but less than five.



Unit Set is a finite set that has only one element. Examples are: C={0 } o o Set D is the set consisting of the number that is neither prime nor composite.



Infinite Set is a set where the number of elements is unlimited or not countable. Examples : o E={…,−6 ,−4 ,−2,0, 2, 4, 6, … } o Set F is the set of positive integers



Empty or null set is a set that has no elements. Examples are : G={} or G=∅ o o Set H is the set of months that start with the letter Z.



Cardinality or cardinal number of a set is the number of elements in it . For example , the cardinal number of set A (describe above) is denoted by n( A ) =5 . Take note that in finding for the cardinality of a set elements that are listed more than once are counted only once. For example the set T ={3, 3, 4,7,8 } has a cardinality of 4 and is denoted as n ( T ) =4



Equivalent Sets are sets that have the same number of elements. In other words, they have the same a cardinality. For example , set C and D are equivalent sets denoted by C D since they have the one element each ; that is n ( C ) =n(D ) . However , Set A and Set B are not equivalent because Set A has five elements and Set B has only four elements.



Equal sets are sets that have exactly the same elements. Examples are : o If V ={a , e , i ,o ,u } and W ={i, e , o , a ,u } , then Sets V denoted by V =W . PANGASINAN STATE UNIVERSITY

and

W

are equal

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

o If N={ x∨x ∈ Z }

Module 2 : Mathematical Language and Symbol

and L={… ,−3,− 2,−1, 0,1, 2, 3,… } , then

N=L .



The symbol ∈ is used to indicate that an element belongs to a set ;while is used to indicate that an element does not belong to a set .Example are : o Given V ={a , e , i ,o ,u } , we say that a is an element of V ” or in symbols, a ∈ V . o Given N={ x∨x ∈ Z } , we say that “ 0.5 is not an elementof Set N or in symbols , 0.5 ∉ N.



The symbol ⊂ is used to indicate that set is a proper subset of another set. By proper subset , in given two sets A and B , every element of Set A is also an element of Set B , but not all elements of Set B are elements of Set B are in Set A . Such a relation between sets is denoted by A ⊂ B and is read as “ A is a proper subset of B .” . It is ⊂ is the one with more elements . important to note that the set on the right of

On the other hand , the symbol ⊄ is used to indicate that a set is not a subset of another set ; meaning not all elements of the first set are also element of the second set.

 The symbol ⊆ is used to indicate that equal sets are subset of one another . Suppose we are given two sets , V ={a , e , i ,o ,u } and W ={i, e , o , a ,u } . Since Sets V and W are equal sets, we say that V is a subset of W , and conversely, W is a subset of V . In set notation , we state V ⊆ W and W ⊆V . 

Power set is the set composed of all the subsets of a given set. For example , the power set of set A={2, 4, 6 } denoted as P ( A ) , is {∅ , { 2} , { 4} , { 6} ,{ 2,4 } , { 2,6} , { 4,6} , {2,4,6 } }. Do note that an empty set is a subset of every set and every set is a subset of itself.

 Sets that have common elements are called joint sets ; while those that do not have common elements are called disjoint sets.

Your turn evaluate if

Fill in the table below with corresponding notation of statement and what it states is true or false.

K={ 1,2, 3, 4, 5 }

L={ 2 4,5 }

Notation L⊂ K

M ={ 0,4,7,10,15 } Statement

N={ 2,4,5,3,1 } True/False

L⊄ M N⊆ K ∅⊂ N

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

Your turn

Do you notice any relation between the number of elements in a set and number of elements in its power set ; that is the number of subsets (proper subsets plus its equal set)?What seems to be the pattern? Can you come up with a formula?Go and investigate! Put your observations in the table . Number of Elements in a Set

Number of Subsets

0 1

2 3

8 4 5 6 n

3.2 Set Operations In this section we will discuss the set operations such as : Union, Intersection, Difference, Complementation, and Cartesian Product.  The union of two sets X a nd Y is the set composed of elements that belong to either Set X a nd or Set Y or both sets , and is denoted by X ∪Y which read as X union Y .

Example 3

then Find

If Set A is the set composed of months starting with letter J. Set B is the set composed of months with exactly five letters. Set C is the set composed of months starting with letter M.

A=¿ {January,June, July} A ∪B

, A ∪C

,

B=¿ {March, April} B∪C

C=¿ {March, May}

.

Solution A ∪ B={ January,March, April,June, July } A ∪C={ January,March, May,June, July } B ∪ C=¿ { March, April, May} In “March”, the common element of Sets B and C is written only once in

Your turn

B∪C .

Perform the indicated operation. Use the same sets given in example 3.

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

a. C ∪ B b. A ∪ B ∪C  The intersection of two sets X a nd Y is the set composed of elements that belong to either Set X a nd or Set Y or both sets , and is denoted by X ∩Y which read as X intersection Y .

then July}

If

D is the set of single-syllable months. E is the set of months with y as the letters. F is the set of months with four letters or less. E={ January, February , July} D=¿ {March, May, June }

Example 4

Find D∩ E

,

E∩F

F={ May, June,

, D∩ F

Solution D∩ E=∅ E ∩ F=¿ {July} D ∩ F=¿ {May, June} Set D and E are disjoint sets so their intersection is empty set.

Your turn

Perform the indicated operation. Use the same sets given in example 5.

F∩D a. b. D∩ E∩ F c. ( D ∩ E)∪ ( E ∩ F)

 The difference of two sets X a nd Y is the set of elements in set X , but its common elements with Set of Y is taken out and is denoted by X −Y which is read as ” X minus Y . “ X −Y can be understood also as the set composed of the elements of X with the elements of its intersection with Y removed or in symbols, X −Y =X − ( X ∩Y ) .

Example 5

Given A=¿ {January , June , July} Find

D=¿ {March, May, June}

F=¿ {May , June, July}

A−D , D− A , A−F , F − A

Solution A−D =¿ {January, July} D− A=¿ {March , May} PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

A−F=¿ {January} F−A=¿ {May} Note : A−D ≠ D − A and

A−F ≠ F − A .

Perform the following using the same sets given in Example 5.

Your turn a. D−F b. F−D c. ( A ∪ D ) −F d. ( F ∩ D) − A e. ( A ∪ F )− ( A ∩ D )

 The complement of a Set X relative to a universal set U is the set of elements in U that are not in X and is denoted by X ' (read as” X prime”). The universal set is the totality of all elements that are included under a defined condition. X ' can be understood as the set that is composed of all elements of U with its common elements with X taken out , or in symbols , X ' = U − X .

Example 6

Find

A

'

and

If

B

U={− 5,−4,−3,−2,−1,0, 1, 2,3, 4, 5 } A={0,1, 2,3, 4, 5 } B={− 4,−2, 0, 2, 4 }

'

Solution A ' ={−5,−4,−3,−2,−1 } B ' ={−5,−3,−1,1, 3,5 }

Your turn a. b. ( A ∩B)' ' c. ( A ∪ B )− A ' d. B ∩(A − B )

Perform the indicated set operations. ( A ∪ B)'

 The Cartesian product of two sets X and Y is the set of all possible pairs of elements and is denoted by X ×Y (read as “the Cartesian product of X and Y ”). Each pair of elements is called an ordered pair ( x , y) , where the first element x is an element of the first set X ; that is , x ∈ X ; the second element y is an element of the second set Y ; that is y ∈ Y . PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 2 : Mathematical Language and Symbol

If M ={ 0,1 }

Example 7 Find M × N

,

N×M ,

and and

N={ 1,2 } M×M<...