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I. Introduction
Different cultures used different languages. And every different language has
different symbols, and every symbols of the different cultures has their own unique
meaning, and it can be represented in different ways. like for example to represent a
number “four”. It can be written in symbols as IV, kwatro, IIII, and others.
Nevertheless, mathematical language and symbols is a channel for communication
now a day, thought as unified in the universe.
Trivia
GREATER THAN AND LESS THAN
SYMBOLS
The symbols > for “ is greater than”
and < for “ is less than “ were the
inventions of Thomas Harriot. An
English mathematician and astronomer.
These symbols were found in his
algebra book published in 1631, ten
years after his death. At the same time,
many mathematicians were ready using
the symbols.
The symbols ≥ and ≤ were the
inventions of Pierre Bonguer in 1734.
THE EQUALITY SYMBOLS
The Englishman Robert Recorde is
Credited with the invention of the equal
sign. He wrote a mathematics book
called “Wholesome of white.” Which
was published in 1557. In that book he
used the symbol “ “ to avoid
repeating “ is equal to.” The reason why
he chooses a pair of line segments of
the same length is because he thought
that no two things are equal. Eventually,
the segments shortened until it became
“=”. Later, another English
mathematician, Thomas Harriot helped
popularize the equality sign and also
invented the “>” and the “ < “ symbols
AGE DOESN`T MATTER
Maria is half the age of his boyfriend
and the sum of the figures in the
boyfriend’s age is half the sum of those
in the girlfriend’s age. How old his
boyfriend?
Ans. Maria is 15 years of age
Boyfriend is 30 years old.
THE POWER OF 2
The power of 2 ( 2
0
1
2
.....) is a
strong set of numbers. Any natural
number is generated by the element or
sum of elements of the power of 2.
For example,
0
1
0
1
2
2
2
2
II. Chapter Discussion
A. Learning Objective
At the end of this section, the student is expected to:
Define and illustrate variable.
Classify whether the statement is expression or sentence
CHAPTER 2
MATHEMATICAL LANGUAGE
AND SYMBOLS
Translate each verbal phrase to mathematical phrase
Translate each verbal sentences to mathematical sentences.
State, define and illustrates the four basic concepts: sets, binary
operations and relations and functions.
Define relation and function.
Identify function from a given relation.
Identify function from a given mapping diagram.
Identify function from a given graph or equation
B. Discussion and Supplementary activities
Lesson 2.1 Variable
Preliminary Activity: Remember me
Answer what is ask:
- Is there a number with the following property: doubling it and adding 3 gives
the same result as squaring it?
- Are there two numbers with the property that the sum of their squares equals
the square of their sum?
Analysis of the Activity
- What does the statements mean?
- What will represent the ambiguous words in the statements?
- How can we rewrite the statements into mathematical form?
Presentation of the lesson
Variable – is a symbolic name associated with an object whose associated value
may be changed. The variable will help us determine the objects or numbers that we
are looking for.
Example 1. How can we use variable to rewrite the statement below formally?
Is there a number with the following property: doubling it and adding 3 gives the
same result as squaring it?
In this statement we can use variable to replace the potentially vague word “it”.
Is there a number with the following property that 2x + 3 = x
2
or
Is there a number with the following property that 2n + 3 = n
2
Using a variable allows us to give temporary value to what we are looking for so that
we can perform computations to discover the possible values. To emphasize the role
of the variable as a placeholder, we might write the following:
Is there a number with the property that 2 • + 3 =
2
The box can also help us fill in the variety of different values that will make the two
side equal or not.
- The cube root of any negative real number is positive.
a. Given any negative real number r, the cube root is ____.
b. For any real number r, if r is ____ then ____.
c. If a real number r _____, then ____.
B. Rewrite the following statements less formally, without using variables. Determine
whether the statements are true or false.
a. There are real numbers s and r with the property that s + r < s – r.
b. There is a real number y such that y
2
< y.
c. For all positive integers u, u
2
≥ u.
d. For all real numbers a and b, |a + b| ≤ |a| + |b|.
Reflection
Write a journal using the following guide questions:
- What is variable?
- What is it for?
- What is it about?
- How is it done?
Lesson 2. 2 Expressions vs. Sentences
Preliminary Activity : Getting to know you
Expression or sentence?
- Ah! Ah! Ah! Ah!
- How are you Ah Ah?
- Shit,, I am not Ah Ah.
- I am Ahagony.
- Ahagony, Is it your Mother?
- Oh,! Yes.
- Mother, one plus two?
- One plus two is equal 5.
- Objection, one plus two is equal three.
- Are you now happy?
Analysis of the Activity
a. What have you notice your activity?
b. Did you find it easy in knowing whether it expression or sentence?
c. When can you say that the statement is expression? A sentence?
d. What is the difference between them?
Presentation of the lesson
Expression
An expression is the mathematical analogue of an English noun; it is a correct
arrangement of mathematical symbols used to represent a mathematical object of
interest.
Examples.
- Any real numbers
- x + 5
- Oh! MJ
- The sum of x and y
- Eight divided two
They are all expressions because each given information does not express a
complete thought.
Sentence
A mathematical sentence is the analogue of an English sentence; it is a correct
arrangement of mathematical symbols that states a complete thought.
Examples.
- Michael is dancing
- Four is a natural number.
- The sum of x and y is 12.
- ¾ is a fraction.
- 3 is an odd number.
All the above examples are mathematical sentence because each statement states a
complete thought.
English phrase Mathematical expression
- The product of a and b ab
- The sum of ten and four 10 + 4
- The difference of three and
two
- Twice the sum of x and y 2( x + y)
English Sentence Mathematical Sentence
- Two times the difference of y
and three is four.
2( y – 3 ) = 4
- Seven is the sum of a and b 7 = a + b
- Ten is the product of five
and x.
10 = 5x
- Five is the square root of
twenty five
Seat work
I. Directions. Write (ME) if Mathematical expressions and MS for Mathematical
Sentence?
- x
- ( a + b)( a - b )
- 5 = x + 7
- x > y
5. Y = 8
II. Translate each of the following below into mathematical sentences.
- The sum of x and y is nine.
- The square root of sixty-four is eight.
- The product of a and b is forty-two.
- The quotient of x and y is seven.
- The product of x and the sum of a and b is 34.
- The months that begins with M are March and May.
- The months that ends with ber are September, November and December.
- August A where A is the set of months between Independence day and
national heroes day.
Analysis of the Activity
- How can you describe your activity?
- How did you know that the statement was true? False?
- How can you define set? A subset?
Presentation of the lesson
SET
A set is a collection of objects. For example, a deck of cards, every student enrolled
in Math 1, the collection of all even integers, these are all examples of sets of things.
Each object in a set is an element of that set. The two of diamonds is an element of
the set consisting of a deck of cards, one particular student is an element of the set
of all students enrolled in Math 1, the number 4 is an element of the set of even
integers. We often use capital letters such as A to denote sets, and lower case
letters such as a to denote the elements.
There is a set of natural numbers designated by the symbol ℕ. Remember that a
natural number is any positive, whole number. This set is ℕ = {1, 2, 3, …}
The next standard set is for integers. Integers are any whole number, whether it is
positive or negative. So this set is ℤ = {…, - 2, - 1, 0, 1, 2,…}
Then there is the standard set of all rational numbers. A rational number is a number
that can be in the form p/q where p and q are integers and q is not equal to zero. So
Next is the set of all real numbers. A real number is any rational or irrational number
that can be placed on a number line like 9/4 or even π. The symbol for the real
numbers set is ℝ. So we would write the set like ℝ = {3.14…, 2/5, 8, …}
Symbols
“ ∣ “means “such that”. This one is a little odd because it contains instructions for the
set.
E.g., A = {x
2
∣ x = ℕ}. The set would be A = {1, 4, 9, 16,25,36, …}
∈ is the symbol for element,includes or “is in”. For example A ∈ (-1, 1), so the set is
A = {-1, 0, 1}.
Definition 1. Given a set A, if u is an element of A
we write u ∈ A.
If the element u is not in the set A we write u ∉A. Some sets that you may have
encountered in mathematics courses before are:
The integers Z
The even integers 2Z
The set of rational numbers Q
The set of real numbers R. We can now practice using our element notation:
Example 1. We have 4 ∈ 2Z.
Example 2. 16∈Z
Example 3. √3∉ Q
So far, we have been defining sets by describing them in words. We can also specify
some sets by listing their elements.
For example, define the set T by writing T ={a,b,c,d,e}. When defining a set by listing,
always use the brackets {,}.
Another set that we can define by listing is the set of natural numbers N
={0,1,2,3,4,···} , where we have indicated a general pattern (hopefully easily
recognized!) by writing ···. Many sets cannot be listed so easily (or at all for that
matter), and in many of these cases it is convenient to use a rule to specify a set.
For example, suppose we want to define a set S that consists of all real numbers
between −1 and 1, inclusive. We use the notation
S ={x|x ∈ R and−1≤x≤1}.
We read the above as “S equals the set of all x such that x is a real number and x is
greater than or equal to −1, and less than or equal to 1.”
What happens if someone specifies a set by a rule like “x is a negative integer
greater than 1000”? What should we do? There are no numbers that are negative
and greater than 1000. We allow examples of rules of this kind, and make the
following definition:
Empty set
The empty set is the set with no elements, and is denoted by the symbol φ, or by {}.
Example 1. { } is not an empty since it contains one element.
Example 2. {x/x is a natural number between 3 and 4}, the given set is empty
because there are no natural numbers between 3 and 4.
Three ways in designating a set.
1. Roster Method- is a method in which the elements are separated by commas
and enclosed by braces
Example 1. Write the set of natural numbers less than 8
Answer. {1,2,3,4,5,6,7}
Example 2.Write the set of odd natural numbers greater than 7.
Answer. {9,11, 13,15…..}
Example 3.Write the set of even numbers between 2 and 10.
Answer. {4, 6, 8}
2.3.2 Binary Operations
Preliminary Activity. Complete me!
Give the correct answer of the following:
1. 5(2) = _________
2. 3(2 + 3) = ________
3. (5 + 2) + 3 = _________
4. 5• 0 = _________
5. 3/4(4/3) = _________
Analysis of the Activity
- Did you find difficulty in getting the correct answer?
- What strategies did you apply in getting the correct answer?
- Did you remember the different properties of real numbers? If yes, what are
those?
- Do you believe that adding two integers the result is always integer?
- Do you believe that dividing integers the result is always integer?
Presentation of the Lesson
Addition and multiplication are the wo basic operations to be used in real numbers.
These two operations is called a binary operation because it takes any two real
numbers as arguments to have another third real number.
In addition, we have a + b, where a and b are called terms , example (3 + 1 = 4)
while in multiplication ab, where a and b are called factors. Example ( 3 • 2 = 6)
Properties of Binary Operations
A. Closure of Binary Operations. The product and the sum of any two real
numbers is also a real number.
In addition, a + b ℝ , for any a, b ℝ
Examples
- Adding two integers the sum is always an integer.
- Adding two real numbers the result is always a real number.
- Adding two natural numbers the result is always a natural number.
The set of integers, natural numbers is always closed under addition.
In multiplication, ab ℝ , for any a, b, ℝ
Examples
- Multiplying two integers the product is always an integer.
- Multiplying two real numbers the result is always a real number.
- Multiplying two natural numbers the result is always a natural number
B. Commutative of Binary Operations. Interchanging the position of two real
numbers. The sum and the product of any two real numbers does not affect.
In addition, Ɐa, b ϵ ℝ, a + b = b + a
Examples
- 6 + a = a + 6
In Multiplication, Ɐa, b ϵ ℝ, a • b = b • a
Examples
- 2a • 3b = 3b • 2a
C. Associative of Binary Operations. The grouping of any given three real
numbers does not affect the sum and the product.
In addition, Ɐa, b, c ϵ ℝ , (a + b) + c = b + ( a + c).
Examples
- (6 + a) + 2 = a + ( 6 + 2)
In Multiplication, Ɐa, b, c ϵ ℝ , (a • b)• c = b • (a • c)
Examples
- (2a • 3b )• 5= 3b • (2a • 5)
D. Distributive of Binary operations. Ɐa ,b, c ϵ ℝ , c (a ± b) = ac ± bc
Examples.
- c (2 + 3 ) = 2c + 3c = 5c
E. Identity Elements of Binary operations. Ɐa ℝ , a + I = a and a • I = a
Examples
1. 2 + 0 = 0 + 2 = 2 for addition the identity ( I ) element is 0.
4. - 5 • 1 = 1 • - 5 = - 5 for multiplication the identity ( I ) element is 1.
F. Inverse of Binary operations. Ɐa ℝ , a +( - a ) = O and a •
1
𝑎
Examples
1
5
1
− 2
Seat work
A. Write the word always, sometimes, or never in the blank to make a true
statement.
- The sum of two positive numbers is___________ a positive number.
- The sum of two negative numbers is ____________a negative number.
- The difference of two natural numbers is _________a natural number.
- The set A = {-2 ,0 ,2}is ___________closed under multiplication.
- The set A = {-1 ,0 ,1}is ___________closed under multiplication.
2.3.4 Relations and Functions
Preliminary Activity. Give me a pair.
Using the table below. Express or write in an ordered pair (x, y) and give the set of
domain and range.
X Y
Father Mother
Son Daughter
Mom Dad
Grandma Grandpa
Teacher Student
Husband Wife
Boyfriend Girlfriend
Uncle Auntie
Analysis of the Activity
- In your activity, what is the value of the domain?
- What is the value of the range?
- {(Father, Mother), (Son, Daughter), (Mom, Dad} is this a relation? or a
function?
- If a relation, what makes this a relation?
- If function, Why?
- Is a relation a function?
Presentation of the lesson
In relations and functions, the pairs in the table are “ordered” which means one
comes first and the other comes second.
A relation is just a relationship between set of information. And it is an any set of
ordered pairs. A relation may be viewed as ordered pairs, mapping design, table,
equation, or written in sentences.
A function is a relation in which no two ordered pairs have the same first element. A
function associates each element in its domain with one element in its range.
A function is a particular kind of relation between sets. A function takes every
element x in a starting set, called the domain, and tells us how to assign it to exactly
one element y in an ending set, called the range. Remember that all functions are
relations and not all relations are functions****.
A. ORDERED PAIR RELATION
Testing Relations to see if they are Functions
We make a “mapping table”. We do this as follows:
- List all the x – value on the left.
- At each x – value, draw an arrow, one arrow pointing to each y – value it has.
- If you see a situation where an x – value has two or more arrows branching to
y – values, then it is not a function.
Example: Which of the following ordered pairs represent function?
A = {(1, 5), (2. 5), (3, 5), (4, 5)}
B = {(1, 8), (2. - 9), (3, 7), (3, 12)}
C = {(2, 4), (1. - 5), (4, 10), (-3, - 87)}
A B C
Input Output Input Output Input Output
Answer: Tables A and C are functions because for one input there is a
corresponding one value of output, however table B is not a function because input x
= 3 has two outputs which is 7 and 12.
B. MAPPING DIAGRAM RELATION
Testing Relations to see if they are Functions
Looking at the “mapping diagram”. If you see a situation where an x – value has two
or more arrows branching to y – values, then it is not a function.
A B
Answer: In A, y is not a function of x (x = - 2 has a multiple outputs), however in B, y
is a function of x , because x has only one output y.
C. GRAPHS
Vertical Line Test for Graphs
- If all vertical lines intersect the graph of a relation in at most one point, the
relation is also a function, one and only one output exists for each input.
- If any vertical line intersects the graphs of a relation at more than one point, the
relation fails the test and is NOT a function. More than one value exists for some
(or all) input values.
2
2
y ± 3
±2√ 2 ±√ 5
x 0 ± 1 ± 2 ± 3
Seat work
A. Which of the following sets of ordered pairs represent functions?
A = {(0, - 2), (1. 4), (-3, 3), (5, 0)}
B = {(-4, 0), (2. - 3), (2, - 5), (3, 12)}
C = {(-5, 1), (2, 1), (-3, 1), (0, - 1)}
D = {(3, - 4), (3. - 2), (0, 1), (2, - 1)}
E = {(1, 8)}
B. Graph the given equation and identify if the equation is a function.
- y = 2x + 4
- y = (𝑥 − 2 )
2
- y = 𝑥
3
2
- x =
2
2
2
Evaluation
A. Which of the following mapping diagram shown represent functions?
A B
C D E
B. Using the vertical line test which of the following graphs shown represent
function?
A B C D E F
SUMMARY
This lesson was about Mathematical languages and symbols. The lesson provided
you to familiar the different mathematical variables and symbols which can be use in
solving mathematical problems, You were also given an opportunity to learn
expressions vs, sentences, the four basic concepts such as sets, binary operation,
relations and functions.
Chapter Test.
I. Determine whether the statement is true or false.
- {-2, - 3, - 5}{a, b, c, }
- { 0} is an empty set
- 0.5 ɛ {1/2, 2/3, 3/4}
- {2, 3, 4} = {natural numbers between 1 and 5}
- {x/x A and x∉ B} is read “the set of all x such that x is an element of A and x
is not an element of B.”
- 1 + 0 = - 1 + [7 + (-7)] is an associative binary.
II. Translate each of the following sentences into mathematical sentences.
- The sum of five and nine is twelve.
- 7 is an integer.
- 6 is greater than seven.
- 10 is the principal square root of 100.
- The square of a number x is twenty-five.
- Ten is an even number.
- The product of ¼ and 4 is one.
- Ten is a multiple of five.
- Four is added by five the result is nine.
- X is greater than or equal six.
III. Classify whether the given sets below is finite or infinite
1. X = {x/x ℕ and x 200}
2. A= { x/x ℕ and x < 3}
3. B= { x/x ℕ and x is between 100 and 200}
_4. Set W is the set of faculty in CAS department
- Set N is the set of all possible telephone number_
IV. Classify whether the given pair of sets below is neither, equal or equivalent.
_1. {{}} and {}
- {1,2,3} and {3,2,1}
- {Pedro, Juan, Jhonny} and {Maria, Petra, Josepha}
- {313} and {3,1,3}
- {3,5,6}and {3,3,5,6}_
References:
Sobecki, Dave. 2018. Math in our World. Fourth Edition.
Stewart, Ian_. Nature’s Numbers_
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Publishing Inc.
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with Recreational mathematics
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Prepared by:
JOEL R. SINTOS, RANDY GABON, Dr RENE BRAVO NOVILLA
