QUARTER 3:
MATHEMATICAL
SYSTEM
OBJECTIVES:
1. Define a mathematical system.
2.Illustrates the components
mathematical system.
of
a
3.State the relationships among the
components of a mathematical
System.
INTRODUCTION
When you were in Grade 7, you learned
that points, lines and planes are classified as
undefined terms in Geometry. These undefined
terms are considered the first component of a
mathematical system. The knowledge and skills
you acquired are very important for you to
understand
its
relationship
to
other
components of a mathematical system.
An undefined term is a word/term left
undefined in the system. It is a term already in
its basic form which can be illustrated through
concrete examples. Undefined terms are used
to define or explain more complex terms or
concepts. In Geometry, the undefined terms are
points, lines and planes.
Consider the figure below. Its edges represent lines, its
corners represent points and the flat surface on top, sides
and bottom represent planes.
Line
Plane
Point
Activity 1. Models
of Points, Lines and
Directions: State whether the following objects
Planes
represent a point, a line or a plane.
Point
_____________
1. Star in the sky.
Plane
_____________
2. Cover of a book.
Point
_____________
3. A grain of rice.
Line
_____________
4. Cable wire.
Plane
_____________
5. Laptop screen.
Activity 1. Models
of Points, Lines and
Directions: State whether the following objects
Planes
represent a point, a line or a plane.
Line
_____________
6. A clothesline.
Line
_____________
7. Curtain rod.
Point
_____________
8. Tip of a pen.
Line
_____________
9. Edge of a table.
Point
_____________
10. Tip of a needle.
Activity 2. ENGAGE!
Write T if the statement is correct and F if the statement is false.
State your reasons for your answers, illustrate if necessary.
F
________1.
Only one line can pass through a single point.
Reason: Infinite number of lines can pass through a single
point. This can be illustrated using the following figure.
Activity 2. ENGAGE!
Write T if the statement is correct and F if the statement is false.
State your reasons for your answers, illustrate if necessary.
F
________2.
There are infinite number of lines which pass
through two distinct points.
Reason: Since through two distinct points, only one line can
pass. In the following figure, it can be seen that there is only one
single line that can pass through two distinct points P and Q.
Activity 2. ENGAGE!
Write T if the statement is correct and F if the statement is false.
State your reasons for your answers, illustrate if necessary.
T
________3.
A line segment can be extended indefinitely on both
the sides.
Reason: A terminated line can be produced infinitely on both
the sides.
Activity 2. ENGAGE!
Write T if the statement is correct and F if the statement is false.
State your reasons for your answers, illustrate if necessary.
T
_________4.
If two circles are equal, then their radii are equal.
Reason: Two circles of equal area will have the same radius
from the relation area = 𝜋𝑟 2
Activity 2. ENGAGE!
Write T if the statement is correct and F if the statement is false.
State your reasons for your answers, illustrate if necessary.
T
_________5.
Using the figure below, if AB = PQ and PQ = XY,
then AB = XY.
Reason: From the axiom that if two things are, separately,
equal to a third thing, then they are equal to each other.
In the 3rd century, a Greek mathematician
named Euclid created a mathematical system that
we now called Euclidian geometry, or plane
geometry. Euclid organized most of the known
mathematics of his time so that virtually all
theorems were proved from small collection of
definitions and axioms, and thus axiomatic
method is born.
Today, the axiomatic method is the
distinctive structure of mathematics
(and much of science). No mathematical
claim is accepted unless it can be proved
from basic axioms. The Euclidian
geometry mathematic system is an
axiomatic system.
In
consists
an axiomatic system
four essential components:
essence,
of
undefined terms, clearly stated definitions
(defined terms), a list of intuitive
assumptions called postulates (properties)
and theorems, or new geometric statements
that can be validated.
A. Undefined terms
You have learned from the previous level
that Geometry classifies points, lines, planes, and
space as undefined terms because it is easier to
understand what they are from description of their
properties, than to attempt to give them a precise
definition. Undefined terms play a vital role in
giving the definition of the defined terms in
geometry.
B. Defined terms or Definitions
In Geometry, formal definitions are formed
to give clear meaning of a word or phrase.
Defined terms or definitions have formal
definitions. In other words, defined terms are
clarified using undefined terms or other
previously defined terms. Among the definitions
are:
A mathematical system is made up of a
vocabulary and a collection of statements in an
accepted framework of reasoning. The vocabulary
of a mathematical system is made up of undefined
terms and defined terms while the statements of
mathematical system are postulates/axioms and
theorems.
C. Axioms/Postulates
Many statements which are self-evident and
can be accepted without any question are called
axioms/postulates. These statements make used of
undefined terms and definitions. Axioms and
postulates are essentially the same thing:
mathematical truths that are accepted without proof.
Both axioms and postulates serve as the basis of
mathematical proofs to establish theorems. Euclid’s
five main postulates/axioms “obvious universal
truths” were:
Deductive reasoning is the process of
reasoning that begins with a conjecture and
states known facts to support it with the goal to
prove it to be true. When you use the deductive
thinking method effectively, you can say
“therefore” the conclusion, or conjecture, is true
with certainty. If your facts are valid, then your
conclusions will be valid.
D. Theorems
Theorems are established statements using
undefined terms, defined terms, postulates, or
conjectures. Theorems are statements proven based on
strict logical proof. The process of showing a theorem to
be correct is called proof. Every mathematical theorem
began as a conjecture or postulate before they were
tested and accepted as proven mathematical facts. Once
a theorem has been proven, it becomes an accepted tool
in the proof of any other theorems.
Theorems are usually divided into a
hypothesis (what is given) and a conclusion
(what needs to be proven). It is written in
conditional form or if-then form. Every
conditional statement has two parts:
hypothesis, denoted by p and a conclusion
denoted by q. In symbols, “If p, then q” is
written as p q.
Identifying the hypothesis and
conclusion.
If it is an even number, then it is divisible by 2.
Hypothesis: It is an even number.
Conclusion: It is divisible by 2.
Writing a Conditional
Statement
1. A rectangle has four right angles.
Conditional: If a figure is a rectangle, then it
has four right angles.
2. It is 9:30 am, a daytime.
Conditional: If it is a 9:30 am, then it is a
daytime.
A conditional statement can have a truth value
of true or false. To show that a conditional is true,
show that every time the hypothesis is true, the
conclusion is also true. You must construct a logical
argument using reasons (can be a definition, an
axiom or property, a postulate or a theorem). To
show that a conditional statement is false, you only
need to find only one counterexample in which the
hypothesis is true and the conclusion is false.
Finding a Counterexample.
If it is February, then there are only 28 days in
the month.
To show that this conditional is false, you need to
find counterexample that makes the hypothesis true,
and the conclusion false. February in the year 2020 is a
counterexample. Because 2020 is a leap year, the month
of February has 29 days. Therefore, the statement is
false.
Conjectures are often confused
with postulates. Conjectures are
conclusions we make based on things
that we observe. While conjectures
need to be proven before they are
accepted, postulates are given and need
no proof.
The converse of a conditional switches the
hypothesis and the conclusion.
Writing the converse of a
conditional
Conditional: If two lines intersects to form right
angles, then they are perpendicular.
Converse: If two lines are perpendicular, then they
intersect to form right angles.
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for
LISTENING
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