Reference No: KLL-FO-ACAD-000 | Effectivity Date: August 3, 2020 | Revisions No.

: 00

VISION MISSION
A center of human development committed to the pursuit of wisdom, truth, Establish and maintain an academic environment promoting the pursuit of
justice, pride, dignity, and local/global competitiveness via a quality but excellence and the total development of its students as human beings,
affordable education for all qualified clients. with fear of God and love of country and fellowmen.

GOALS
Kolehiyo ng Lungsod ng Lipa aims to:
1. foster the spiritual, intellectual, social, moral, and creative life of its client via affordable but quality tertiary education;
2. provide the clients with reach and substantial, relevant, wide range of academic disciplines, expose them to varied curricular and co-curricular
experiences which nurture and enhance their personal dedications and commitments to social, moral, cultural, and economic transformations.
3. work with the government and the community and the pursuit of achieving national developmental goals; and
4. develop deserving and qualified clients with different skills of life existence and prepare them for local and global competitiveness

MODULE
FIRST Semester, AY 2020-2021

I. COURSE : GE 104 – Mathematics in the Modern World

II. COURSE DESCRIPTION:

The nature of mathematics, appreciationof its practical, intellectual, and aesthetics
dimensions, and application of mathematical tools in daily life.
This course begins with an introduction to the nature of mathematics as an
exploration of patterns (in nature and the environment) and as an application of inductive and deductive
reasoning. By exploring these topics, students are encourage to go beyond the typical understanding of
mathematics as merely a bunch of formulas, but as source of aesthetics in patterns of nature, for
example, and rich language in itself (and of science) governed by logic and reasoning.
The course proceeds to survey ways in which mathematics provides a tool for
understanding and dealing with various aspects of present day living such as managing personal
finances, making social choices, appreciating geometric designs, understanding the codes used in data
transmission and security, and dividing limited resources fairly. These aspects will provide opportunities
for actually doing mathematics in a broad range of exercises that bring out the various dimensions of
mathematics as a way of knowning and test the students understanding and capacity.

III. SUBJECT MATTER

SUBJECT MATTER Time-Frame
Chapter 2. Mathematical Language and Symbols October 5 - , 2020
a. Language, Symbols, and Convention of Mathematics
b. Operations on Mathematical Expression
c. Four Basic Concepts of Mathematics
d. Elementary Logic

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IV. COURSE OUTCOME
At the end of the course, the students should be able to:
Knowledge
1. Discuss and argue about the nature of mathematics, what it is, how it is expressed,presented,
and used.
2. Use different types of reasoning to justify statements, arguements made about mathematicIIiis
and mathematical concepts.
3. Discuss the language and symbols of mathematics.
Skills
4. Use a variety of statistical tools to process and manage numerical data.
5. Analyze codes and coding schemes used for identification, privacy, and security purposes.
6. Use mathematics in other areas such as finance, voting, health and medicine, business,
environment, arts and design.
Values
7. Appreciate the nature and uses of mathematics in everyday life.
8. Affirm honesty and integrity in the application of mathematics to various human endeavors.

V. ENGAGEMENT
LESSON OBJECTIVES
1. Discuss the language, symbols, and conventions of mathematics.
2. Explain the nature of mathematics as a language.
3. Perform operations on mathematical expression correctly, its basic concept and
logic.
4. Appreciate that mathematics is a useful language.

1.1 The Language, Symbols,Syntax and Rules of Mathematics

The language of mathematics is the system used by mathematicians to communicate mathematical ideas
among themselves. This language consists of a substrate of some natural language (for example English) using
technical terms and grammatical conventions that are peculiar to mathematical discourse, supplemented by a
highly specialized symbolic notation for mathmatical formulas.

Mathematics as a language has symbols to express a formula or to present a constant. It has syntax to make
the expression well-formed to make the character and symbols clear and valid that do not violate the rules.
Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation,
and grouping to hepl determine order of operations, and other aspects of logical syntax. A mathematical concept is
independent of the symbol chosen to represent it. In short, convention dictates the meaning.

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The language of mathematics (wikipedia) makes it easy to express the kinds of symbols, syntax and rules that
mathematicians like to do and characterized by the following:

a. Precise (able to make very fine distinctions)

Example: The use of mathematical symbol is only done based on its meaning and purpose. Like + means add,
- means subtract, x means mutiply and ÷ means divide.
b. Concise (able to say things briefly)
Example: The long English sentence can be shortened using mathematical symbols. Eight plus two equals ten
which means 8+2=10.
c. Powerful (able to express complex thoughts with relative case)
Example: The application of critical thinking and problem solving skill requires the comprehension, analysis,
and reasoning to obtain the correct solution.

1.2 Writing Mathematical Language as an Expression or a Sentence

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed

according to the rules that depend on the context. It is the correctt arrangement of mathematical symbols used to
represent a mathematical object of interest. An expression does not state a complete thought; it does not make sense
to ask if an expression is true or false.

The most common expression types are numbers, sets, and functions.

5 2+3 10/2 (6-2)+1 1+1+1+1+1

On the other hand, a mathematical sentence is the analoge of an English sentence; it is a correct arrangement of
mathematical symbols that tates a complete thought. Sentences have verbs. In the mathematical sentence ‘3+4=7’ ,
the verb is ‘=’.

A sentence can be (always) true, (always) false, or sometimes true/ sometimes false. For example, the sentence
‘1+2=3’ is true. The sentence ‘1+2=4’ is false.

The sentence ‘x=2’ is sometimes true/sometimes false: it is true when x is 2, and false otherwise. The sentence
‘x+3=3+x’ is (always) true, no matter what number is chosen for x.

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1.3 Mathematical Convention

A mathematical convention is a fact, name, notation, or usage which is generally agreed upon by mathematics. For
instance, the fact that one evaluates multiplication before addition in the expression (2+3)x4 is merely conventional.

The following symbols are commonly used in the order of operations:

SYMBOL MEANING EXAMPLE

+ Add 3+7=10

- Subract 5−2=3

X Multiply 4 ×3=12

÷ Divide 20 ÷ 5=4

/ Divide 20/5 = 4

π pi A=π r
2

∞ infinity ∞ is endless

¿ Equals 1+1=2

≈ Approximately equal to π ≈3.14

≠ Not equal to π≠2

¿≤ less than, less than or equal to 2<3

¿≥ greater than, greater than or equal to 5>1

√❑ Square root (“radical”) √ 4=2

° degrees 20 °

∴ Therefore a=b ∴ b=a

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Comprehending a message is better understood once a person understand how things are said and may know why it
is said. The use of language in mathematics is far from ordinary speech. Ir can be learned but need a lot of efforts like
learning a new dialect or language. The following are the characteristics of the language or mathematics: precise,
concise and powerful.

 Expressions versus Sentences

You learned in your English subjects that expressions do not state a complete thought, but sentences do.
Mathematical sentences state a complete thought. On the other hand, mathematical expressions do not. You
cannot test if it is true or false.

MATHEMATICAL EXPRESSIONS MATHEMATICAL SENTENCES

26.14 -11+7=4
5+2 1-4=-3
X+√2 1∙X=X

 Conventions in the Mathematical Language

The common symbol used for multiplication is x but it can be mistakenly taken as the variable x. There are
instances when the centered dot (∙) is a shorthand to be used for multiplication especially when variables are
involved. If there will be no confusion, the symbol may be dropped.

8 ∙ y = 8y
a ∙ b ∙ c = abc
t ∙ s ∙ 9 = 9st

It is conventional to write the number first before the letters. If in case the letters are more than one, you have
to arrange the letters alphabetically.

Sets are usually represented by uppercase letters like S. The symbol R and N represent the set of real numbers and
the set of natural numbers, respectively. A lowercase letter near the end of the alphabet like x, y, or z represents an
element of the set of real numbers. A lowercase letter nead a middle of the alphabet particularly from i to n may
represent an element of the set of integers

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1.4 Perform Operations on Mathematical Expressions Correctly

In simplifying mathematical expressions, the following order of operations is one of the critical point to
observe. Order of operations is the hierarchy of mathematical operations. It is a set of rules that determines which
operations should be done before or after others. Before we used to have the MDAS, that stands for Multipication,
Division, Addition, and Subtraction. It was change to use PEMDAS which means Parethesis, Exponents, Multipication
and Division, and Addition and Subtraction. But now, most scientific calcuators follow BODMAS, that is Brackets,
Order, Division and Multiplication, and Addition and Subraction.

The order of operations or BODMAS/ PEMDAS is merely a set of rules that prioritize the sequence of operations
starting from the most important to the least important.

Step 1: Do as much as you can to simplify everything inside the parenthesis first.

Step 2: Simplify every exponential number in the numerical expression.

Step 3: Multiply and divide whichever comes first, from left to right.

Step 4: Add and subtract whichever comes first, from left to right.

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Examples:

1. Evaluate: ( 11−5 ) ×2−3+1

Solution:
Remove the parenthesis: 6 ×2−3+1
Multiply: 12−3+ 1
Subtract :9+1
Add: 10

2. Evaluate: 10 ÷ 2+ 12÷ 2× 3

Solution:

Using PEMDAS rule, we need to evaluate the division and multipication before subtraction and addition. It is
recommended that you put in parentheses to remind yourself the order of operation.

From the given, 10 ÷ 2+ 12÷ 2× 3

¿ ( 10 ÷2 )+(12÷ 2 ×3)

¿ 5+18

¿2

1.5 The Four Basic Concepts of Mathematics

1. Set

A set is a collection of well-defined objects that contains no duplicates. The objects in the set are called the elements
of the set. To describe a set, we use { }, and a capital letters to represent it.

Examples: The following are examples of sets:

1. The books in the shelves in a library

2. The bank accounts in a bank

3. The set of natural numbers N = {1,2,3,...}.

4. The integer numbers Z = {... ,-3,-2,-1,0,1,2,3,...}.

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5. The rational numbers is the set of quotients of integers Q={ p/ q : p , q ∈ Z∧q=0 }

The three dots in enumerating the elements of the set are called ellipsis and indicate a continuing pattern. A finite set
contains elements that can be counted and terminates at certain natural number, otherwise, it is infinite set.

Examples:

Set A = {2, 4, 6, 8, 10}

-the set of all even natural numbers less than or equal to 10. The order in which the elements are listed is not
relevant: i.e., the set {2, 4, 6, 8, 10} is the same as the set {8, 4, 2, 10, 6}.

There is exactly one set, the empty set, or null set, ∅ , { } , which has no members at all. A set with only one menber is
called a singleton or singleton set. (“singleton of a”).

Specification of Sets

There are three ways main ways to specify a set:

(1) List Notation/ Roster Method – by listing all its members

- list names of elements of a set, separate them by commas and enclose them in braces:

Examples: 1. {1, 12, 45}

2. { George Washington, Bill Clinton},

3. {a, b, d, m}

4. “Three dot abbreviation”: {1, 2, ... , 100}

(2) Predicate Notation/ Rule Method/ Set Builder Notation – by stating a property of its elements. It has a property that
the members of the set share ( a condition or a predicate which holds for members of this set).

Examples:

1. {x/x is a natural number and x<8} means “the set of all x such that x is a natural number and is less than 8”

2. {x/x is a letter of Russian alphabet}

3. {y/y is a student of Umass and y is older than 25}

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(3) Recursive Rules – by defining a set of rules which generates or defines its members

Examples

1. the set E of even numbers greater than 3:

a. 4∈E

b. if x ∈ E, then x+2 ∈ E

c. nothing else belongs to E. Les)

Equal Sets

- Two sets are equal if they contain exactly the same elements.

Examples:

1. {3, 8, 9} = {9, 8, 3}

2. {6, 7, 7, 7, 7} = {6, 7}

3. {1, 3, 5, 7} ≠ {3, 5}

Equivalent Sets

Two sets are equivalent if they have contain the same number of elements.

Example:

1. Which of the following sets are equivalent?

{∅ , ϑ , β }, {1, 4, 3}, {a, b, c}

Solution: All of the given sets are equivalent. Note that no two of them are equal, but they all have the same number of elements.

Universal Set

A set that contains all the elements considered in a particular situation and denoted by U.

Example: The universal set

a. Suppose we list the digits only.

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Then, U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, since U includes all the digits

b. Suppose we consider the whole numbers

Then U = {0, 1, 2, 3, ... } since U contains all whole numbers.

Subsets

A set A is called a subset of set B if every element of A is also an element of B. A is a subset of B is written A⊑B.

To identify the number of subset 2n , n = no. Of element .

Example: Subsets

1. A = {7, 9} is a subset of B = {6, 9, 7}

2. D = {10, 8, 6} is a subset of G = {10, 8, 6}

 A proper subset that is not equal to the original set, otherwise improper subset.

Example Given {3, 5, 7}, then the proper subsets are { }, {5, 7}, {3, 5}, {3,7}. The improper subset is { 3, 5, 7}

Operation on Sets

Union is an operation for set A and B in which a set is formed that consist of all the elements included in A or B or both denoted
by ∪ as A∪B.

Examples:

1. Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 5, 7}, B = {2, 4, 6, 8} and C = {1, 2}, find the following:

a. A ∪B b . A ∪C c .( A ∪B) ∪C

Solution:

a. A ∪B= {1 , 2, 3 , 4 ,5 , 6 , 7 , 8 }

b. A ∪C={ 1 ,2 , 3 ,5 , 7 }

c. ( A ∪B ) ∪ C={1 , 2 ,3 , 4 , 5 ,6 ,7 , 8 }

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Intersection

- Is the set containing all elements common to both A and B, denoted by ∩.

Example: Given U ={ a , b , c , d , e } , A={ c , d , e } , B= { a ,c , e }∧C={ a }∧D= {e }

Find the following intersections of sets: a . B∩C b . A ∩C c .( A ∩ B)∩ D

Solutions:

a . B∩C= { a } b . A ∩C= { ∅ } c . A ∩ B={ c ,e } , ( A ∩ B ) ∩ D= { e }

Complementation

- Is an operation on a set that must be performed in reference to a universal set, denoted by A’.

Example: Given U ={ a , b , c , d , e } , A={ c , d , e } , find A ' .

Solution: A’ = {a, b}

Functions and Relations

2. Relation

- A relation is rule that pairs each element in one set, called domain, with one or more elements from a
second set called the range. It creates a set of ordered pairs.

Example:

1. Given:

Regular Holidays in the Philippines Month and Date

New Year’s Day January 1

Labor Day May 1

Philippines’ Independence Day June 12

Bonifacio Day November 30

Rizal Day December 30

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 A clearer way to express a relation is to form a set of ordered pairs;

(New Year’s Day, January 1)

(Labor Day, May 1)

(Philippines’ Independence Day, June 12)

(Bonifacio Day, November 30)

(Rizal Day, December 30)

This set describes a relation.

2. {(1,4), (2,5), (3,6)} is a relation. The domain of the relation is the set {1,2,3} and the range is {4,5,6}.

3. Functions

- A rule that pairs each element is one set, called the domain, with exactly one element from a second
set called the range. This means that for each first coordinate, there is exactly one second coordinate
or for every first element of x, there corresponds a unique second element y.

Remember: A one-to-one correspondence and many-to-one correspondence are called functions while
one to many correspondences is not.

Example:

The Function can be represented using the following:

1. Table

Sides 1 3 5 7 9

Perimeter 4 12 20 28 36

2. Ordered Pairs

{(1,4), (3,12), (5,20), (7,28), (9,36)

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 3. Mapping

1 4
3 12
5 20
7 28
9 36

4. Graphing

Using vertical line test, that is, a set points in the plane is the graph of a function if and only if no vertical
line intersects the graph in more than one point.

Not a Function Function

Figure A

A. Binary Operations
A unary operation is for a single number and assigns another number to it. Addition (+), subtraction (-),
multiplication (x)y and division (+) are examples of binary operations. The word "binary" means composition of two
pieces. A binary operation refers to joining two values to create a new one.
Study the following properties of addition and multiplication, as binary operations, on the set of real numbers.
1. Closure Property
Addition: The sum of any two real numbers is also a real number.
Example: 12 + 34 = 46

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 Multiplication: The product of any two real numbers is also a real number.
Example: 7 x 20 = 140
2. Commutative Property
Addition: For any two real numbers x and y, x + y = y + x
Example: 1.5 + 7.8 = 7.8 + 1.5
9.3 = 9.3

Multiplication: The product of any two real numbers is also a real number.
Example: 8 x 5 = 5 x 8
40 40

Applying the commutative property of addition, we may write the equivalent expression of 8m + 7n as 8m +
7n = 7n + 8m. What do you think is the equivalent expression of (12a) • (17b)?
3. Associative Property
Addition: For any two real numbers x, y and z, x + (y + z) = (x + y) + z.
Example: 3 + (10 + 9) = (3 + 10) + 9
3 + 19 = 13 + 9
22 = 22

Multiplication: For any two real numbers x, y and z, x • (y • z) = (x • y) • z.

Example: 3 (9 6) = (3 9) • 6
3 (54) = (27) 6
162 = 162
Using the associative property of multiplication, we may write the equivalent expression of (42') • (16a 2b5 ) as (42c3 ) •
(16a2b5) -- (16a2b5 ) • (42c3 ).

4. Identity Property
Addition: For any real number x, x + 0 = x. The number "0" is called the additive identity.
Example: 78 + O = 78
Multiplication: For any real number x, x • 1 = x. The number "1 " is called the multiplicative identity
Example: 98 • 1 = 98
Using the identity property, we may write an equivalent expression of a mathematical expression by substituting an
expression that is equal to the additive identity or multiplicative identity. Study the examples below.

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 3c 7
Example 1. Write an equivalent expression of by multiplying instead of 1.
5 7

3c 3c
= ∙1
5 5
3c 7
= ∙
5 7
21 c
=
35
21 c 3c
Therefore , and are equivalent expressions.
35 5
Example 2. Write an equivalent expression of 9j 2 - 5 by adding (2m - 2m) instead of O.
9j2—5 = 9j2—5 + O
= 9j2 - 5 + (2m - 2m)
= 9j2 + 2m – 5 - 2m

Therefore, 9j2—5 and = 9j2 + 2m – 5 - 2m are equivalent expressions.

5. Distributive property of Multiplication Over Addition

For any two real numbers x, y and z, x(y + z) = xy + xz.
Using the distributive property, write the equivalent expressions of the following.

1. a( - x + y - z) =____________________________________
2. -5 (71 + 8m + 9n) =____________________________________
3. M(vi – vf) =____________________________________

6. Inverses of Binary Operations

Addition: For any real number x, x + (-x) = 0.
Example: 100 + (-100) = 0

1
Multiplication: For any real number x, x ∙ = 1.
x
1
Example: 98 ∙ =1
89

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1.6 Elementary Logic

According to David W. Kueker (2009), logic is simply defined as the analysis of methods of
reasoning. In studying these methods, logic is interested in the form rather than the content of arguement.
Mathematical Logic is, atleast in tis origins, the study of reasoning as used in mathematics. Mathematical
reasoning is deductive – that is, it consists of drawing (correct) conclusions fromgiven hypotheses. Thus
the basic consept is that of a statement being a logical consequence of some other statements. In
ordinary mathematical English the use of “therefore” customarily indicates that the following stateement is
a consequence of what comes before.

Examples:

1. All men are mortal. Luke is a man. Hence, Luke is mortal

2. All dogs like fish. Cyber is a dog. Hence, Cyber likes fish.

Proposition and Connectives

A proposition (statement) is a sentence that is either true or false (without additional information).

The logical connectives are defined by truth tables (but have English language counterparts).

Logic Math English

Conjunction ∧ And

Disjunction ∨ Or (inclusive)

Negation ∼ Not

Conditional → If...then...

Biconditional ↔ If and only if

A denial is a statement equaivalent to the negation of a statement.

Examples:

1. The negation of P→ Q is ∽ ( P →Q ) .

2. A denial of P→ Qis P∧ ∼ Q

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A tautology is a statement which always true.

Examples: 1. A ∨(B ∧C )↔( A ⋁ B) ∧( A ∨ C) Distributive Law

2. ∼( A ∨ B)↔∽ A ⋀ ∽B

3. P ↔∼ (∼P)

A contradiction is a statement which is always false.

Example: A ∨∼ A ¿ →(B∧ ∼B)is a contradiction.

ACTIVITIES
A. Multiple Choice: Encircle the letter that corresponds to the correct answer.
1. Which is the language system that uses technical terms and grammatical conventions peculiar to
mathematical discourse and is supplemented by a highly specialized symbolic notation for mathematical
formulas?
a. Mathematical Language c. Binary function
b. Set d. Singleton
2. Which is used to express a formula or to represent a constant?
a. Syntax c. Rules
b. Symbols d. Convention
3. Which of the following does NOT belong to the characteristics of the language of mathematics?
a. Symbolic c. Concise
b. Precise d. Powerful
4. Which is a correct arrangement of mathematical symbols and is used to represent a mathematical
object of interest?
a. Mathematical expression c. Rule
b. Relation d. Function

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5. Which is a collection of well-defined objects that contains no duplicates?
a. Function c. Binary
b. Relation d. Set
6. Which sets contains all the elements in a particular situation?
a. Union of sets c. Intersection of sets
b. Univesal sets d. Combination of sets
7. What is a statement that is either TRUE or FALSE?
a. Connective c. Denial
b. Proposition d. Logic
8. Which statement is always TRUE?
a. Proposition c. Tautology
b. Connective d. Denial
9. Which relation is describe as one-to-one correspondence and many-to-one correspondence?
a. Tautology c. Set
b. Logic d. Function
10. Which staement is always FALSE?
a. Tautology c. Set
b. Logic d. Contradiction

B. Solve the following mathematical expression correctly.

1. 4 + ( 1× 5+7 ) +8
2 2

2. 62 + ( 2−8 )+ 1−8

3. ( 6 ×7 ) + 2
2

4. 9−( 52 +7 ) + 2
5.(6+ 8−2)
2
6.(1−4 ) ×2

7. 5+ ( 4 +1 ) +8
2

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8. ( 4 +8 ) +1
2
9.(2 +9+ 1)
10.(3+8−1)
2
45 3 (2 )
11. +
8 ( 5−4 )−3 5−3
13+(−3)2+ 4(−3)+1−[−10−(−6)]
12.
{[4+ 5]÷ [42−3 2(4−3)−8]}+12

C. Find the following.

GIVEN: U ={ 1, 2 , 3 , 4 , 5 ,6 ,7 }

A={ 1, 2 , 3 , 4 , 5 }
B= {2 , 5 ,7 }
C={ 3 , 5 }
1. A ∪ B=¿

2. A ∪ C=¿

3. B∪ C=¿

4. A ∪ B ∪C=¿

5. A ∩ B=¿

6. A ∩C=¿

7. B∩ C=¿

8. A ∩ B∩ C=¿

9. A' =¿

10. B' =¿

11. C ' =¿

12. ( A ∪ B )' =¿

13. ( A ∩ B )' =¿

14. ( B∩ C )' =¿

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 15. ( A ∩C )=¿

D. Directions: Classify each given equation as a mathematical expression (E) or a mathematical sentence (S).

1. A + 9 ______________________
2. b + 0 = b ______________________
t
3. ______________________
100
4. 3.1416 ______________________
5. x + y = y + x ______________________

E. What is the most conventional way to write 5 ∙ 11? Justify your answer.

Answer:
______________________________________________________________________________________________
______________________________________________________________________________________________
______________________________________________________________________________________________

F. Directions: For each of the following expressions, write each in the most conventional way.

1. √3 ∙ x ______________________
2. z∙y∙5 ______________________
3. 8∙y∙x ______________________
4. c2 ∙ a4 ∙ b2 ∙ 3 ______________________
5. 11 ∙ z10 ∙ t ∙ y6 ______________________

VI. OUTPUT RESULTS

Write your answer on the space provided. Keep this module clean and neat.

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VII. EVALUATION
Each question below is given two statements followed by two conclusions numbered I and II. You have to
take the given two statements to be true even if they seem to be at variance from commonly known facts.
Read the conclusion and then decide which of the given conclusion logically follows from the two given
statements, disregarding commonly known facts. Write the letter only.

Select answer:
(A) if only conclusion I follows
(B) If only conclusion II follows
(C) If either I or II follows
(D) If neither I and II follows
(E) If both I and II follows

___________1. Statements: No women can play. Some women teachers are athletes.
Conclusions:
I. Male athletes can play.
II. Some athletes can play
___________2. Statements: No magazine is a cap. All caps are cameras.
Conclusions:
I. No camera is magazine
II. Some cameras are magazines.
___________3. Statements: All huts are mansions. All mansions are temples.
Conclusions:
I. Some temples are huts
II. Some temples are mansions.

Prepared by:

KIMBERLY L. SORUILA, LPT MARK GOLDWIN Q. MOJICA

Instructor I Instructor I

Marawoy, Lipa City, Batangas 4217 | https://www.facebook.com/KLLOfficial/

BERLYN A. FAMILARAN PEDRO B. KATIGBAK
Instructor I Associate Professor I

Checked by:

Department Module Editing Committee

Approved by:

BIBIANA JOCELYN D. CUASAY, Ph.D.

Module Editing Chair

AQUILINO D. ARELLANO, Ph.D., Ed.D.

Vice President for Academic Affairs and Research

Noted by:

MARIO CARMELO A. PESA, CPA

College Administrator

Marawoy, Lipa City, Batangas 4217 | https://www.facebook.com/KLLOfficial/