Module_1_discussion (1)
docx
keyboard_arrow_up
School
Laurel Springs School *
*We aren’t endorsed by this school
Course
PART A
Subject
Mathematics
Date
Nov 24, 2024
Type
docx
Pages
3
Uploaded by BrigadierSeahorse6227
Minsang.Song G9
Application:
To demonstrate that the contrapositive of a conditional statement is equivalent to the converse
of the inverse of the conditional statement, we need to break down each component and
establish their relationships logically. Let's begin by defining the terms involved:
Conditional Statement: A conditional statement is in the form "if P, then Q," where P is the
hypothesis and Q is the conclusion.
Contrapositive: The contrapositive of a conditional statement is "if not Q, then not P."
Converse: The converse of a conditional statement is "if Q, then P."
Inverse: The inverse of a conditional statement is "if not P, then not Q."
We want to show that the contrapositive is equivalent to the converse of the inverse of the
conditional. In other words, we want to prove:
Contrapositive ≡ Converse of Inverse
Let's go through this step by step:
Contrapositive (if not Q, then not P):
If the original conditional statement is "if P, then Q," the contrapositive is "if not Q, then not P."
Inverse (if not P, then not Q):
The inverse of the conditional is "if not P, then not Q."
Now, let's examine the relationships between these statements:
a. If the original conditional statement is true (P → Q), it implies that the contrapositive is also
true (not Q → not P) because denying the conclusion (not Q) means the hypothesis cannot be
true (not P).
b. If the original conditional statement is false, the contrapositive is also false. This is due to the
logical equivalence that "false implies anything" is always true.
c. If the inverse of the conditional statement is true (not P → not Q), it implies that the converse
is also true (Q → P) because affirming the conclusion (Q) means the hypothesis must be true
(P).
d. If the inverse of the conditional statement is false, the converse is also false. This is also due
to the logical equivalence that "false implies anything" is always true.
In summary, the contrapositive and the converse of the inverse are equivalent under both true
and false conditions, as demonstrated in points a, b, c, and d. Therefore, the contrapositive of a
conditional statement is indeed the same as the converse of the inverse of the conditional
statement, making them logically equivalent.
Finding the error(s):
Minsang.Song G9
It is to be noted that the two inaccuracies were that angle 90 was not shown to be produced at
the intersection points of the bisector and that congruent segments were incorrectly indicated.
See the corrected diagrams below.
We can take notice of the following to comprehend better the flaws in the diagram: A
perpendicular bisector cuts a line segment in half and produces a right angle at the point of
intersection.
Congruent segments have an equal number of strokes—specifically, little straight lines to
demonstrate congruency.
Whenever it becomes pertinent to show that two bisected lines are equal, marks resembling an
equal sign (=) must be used to cross the line on both sides of the bisector. The equal sign is
vertical if the line is horizontal, and vice versa. See a sample of the same with the bisection of
DF in the original image.
Discussion Question:
In geometry, definitions are formed by using known words or terms to describe new words.
Three words in geometry are not formally defined. The three undefined terms are point, line,
and plane.
POINT (undefined term)
In geometry, a point has no dimension (actual size). Even if a point is expressed as a point, the
point has no length, width, or thickness. Points are usually named in uppercase letters. Points in
the coordinate plane are named ordered pairs (x,y).
LINE (undefined term)
In geometry, a line has no thickness, but its length extends in one dimension and continues
forever in both directions. The line is shown as a straight line with two arrowheads indicating
that it extends endlessly in two directions. The name of the line is written either as a single
lowercase letter or as two dots with an arrow drawn on the line.
PLANE (undefined term)
In geometry, a plane has no thickness but extends indefinitely in all directions. A flat surface is
usually represented by a shape that looks like a tabletop or wall. It's important to remember
that a floor plan has edges, but a plane has no boundaries. A plane is named by a single letter
(plane m) or by three non-collinear points (plane ABC).
Undefined terms can be combined to define other terms. For example, non-collinear points are
points that do not lie on the same line. A line segment is a portion of a line that includes two
specific points and all the points between them, while a ray is a portion of a line that includes a
specific point, called an endpoint, and all the points that extend indefinitely on one side. of the
endpoint.
Minsang.Song G9
Defined terms can be combined with undefined terms to define more terms. For example, an
angle is a combination of two different rays or line segments that share a single endpoint.
Likewise, a triangle is made up of three non-collinear points and a line segment between them.
Everything else builds on this and adds more information to this foundation. Additions include
all theorems and other "defined" terms such as parallelogram or acute angle.
Reflection:
The course resources and your teacher say that note-taking is helpful in mathematics. As you
went through this module would you agree or disagree with that? What additional advice would
you give to a student who is beginning the course today?"
Clarifying Concepts: Taking notes allows me to write down key concepts and definitions, helping
me understand and clarify my understanding of the material. It's instrumental in mathematics,
where precise definitions are crucial.
Problem-Solving: Writing down steps for solving math problems helps me internalize the
problem-solving process. This can be especially helpful when I encounter similar problems in
the future.
Reference Material: My notes become a valuable reference for studying and reviewing. When
preparing for exams or when working on assignments, well-organized notes save me time and
reduce stress.
Active Learning: Taking notes engages me in active learning. I'm not just passively listening; I'm
actively processing the information, which can enhance my comprehension.
Personalized Learning: My notes reflect my unique way of understanding the material. They
serve as a personalized resource that resonates with my thought process.
What would you do differently in the next module that you didn't do in this module?
I think I didn't review as much as usual in this module. I need to review more than usual, but it's
a shame I couldn't do that. At the same time, I think my grades weren't good either. Next time, I
will work harder and review more.
What new information have you learned from this module? What surprised you about what you
learned? What knowledge from your past courses did you use in this module?
In this module, I finally got the term of reflecting on the x-axis and y-axis. I was not quite sure
about those two. When I got a question about that or another type of question, I felt that I
couldn't solve it as fast enough as the others. Except for that, I think I used my knowledge of
everything that I learned last year. There were some questions that I already did so it was pretty
easy solving on to it.
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help