2.2 Notes Part 1
Summary
TLDRThis educational video script delves into conditional statements, explaining their structure as 'if p then q' where 'p' is the hypothesis and 'q' is the conclusion. It uses examples like 'if today is Thanksgiving, then today is Thursday' to illustrate how to identify hypotheses and conclusions. The script also covers alternative ways to express conditional statements and guides viewers in writing their own using examples of rational numbers, divisibility, and angles. The goal is to enhance understanding of logical structures and their applications.
Takeaways
- 📌 A conditional statement is written in the form 'if p then q', where 'p' is the hypothesis and 'q' is the conclusion.
- 🔍 The hypothesis follows the word 'if', and the conclusion follows the word 'then'.
- 📊 In a Venn diagram, 'p' (hypothesis) is represented in blue and 'q' (conclusion) in red.
- 🌰 Example: 'If today is Thanksgiving Day, then today is Thursday.' Here, 'today is Thanksgiving Day' is the hypothesis, and 'today is Thursday' is the conclusion.
- 🔢 For the statement 'A number is a rational number if it is an integer', the hypothesis is 'a number is an integer' and the conclusion is 'a number is a rational number'.
- 🔄 Sometimes the hypothesis and conclusion can be flipped, as in 'A number is divisible by three if it is divisible by six'.
- 📝 The statement 'if p then q' can also be written as 'p implies q' or 'p only if q', representing the same concept.
- 📐 Using a Venn diagram, the hypothesis is inside the larger circle, and the conclusion is outside but related to the hypothesis.
- 🐦 Example using a Venn diagram: 'If an animal is a blue jay, then it is a bird.' Here, 'an animal is a blue jay' is the hypothesis, and 'it is a bird' is the conclusion.
- 📐 For complementary angles: 'If two angles are complementary, then they are acute.' The hypothesis is 'two angles are complementary', and the conclusion is 'they are acute'.
Q & A
What is a conditional statement?
-A conditional statement is a statement that can be written in the form 'if p then q', where 'p' represents the hypothesis and 'q' represents the conclusion.
What does 'p' stand for in a conditional statement?
-'P' stands for the hypothesis, which is the statement that follows the word 'if'.
What does 'q' represent in a conditional statement?
-'Q' represents the conclusion, which is the statement that follows the word 'then'.
Can you provide an example of a conditional statement from the script?
-Yes, an example from the script is 'If today is Thanksgiving day, then today is Thursday', where 'today is Thanksgiving day' is the hypothesis and 'today is Thursday' is the conclusion.
How are hypothesis and conclusion represented in a Venn diagram?
-In a Venn diagram, the hypothesis (p) is represented in blue and the conclusion (q) is represented in red.
What is another way to write 'if p then q'?
-Alternative ways to write 'if p then q' include 'if p, q', 'p implies q', and 'p only if q'.
In the example 'A number is a rational number if it is an integer', what is the hypothesis?
-The hypothesis is 'a number is an integer'.
In the example 'A number is divisible by three if it is divisible by six', what is the conclusion?
-The conclusion is 'a number is divisible by three'.
What is the significance of identifying the hypothesis and conclusion in a conditional statement?
-Identifying the hypothesis and conclusion helps in understanding the logical relationship between the two parts of the statement and can assist in evaluating the truth of the statement.
How can you write the statement 'An obtuse triangle has exactly one obtuse angle' in if-then form?
-You can write it as 'If a triangle is obtuse, then it has exactly one obtuse angle'.
Using a Venn diagram, how would you represent the statement 'If an animal is a blue jay, then it is a bird'?
-You would place blue jays (hypothesis) within the larger set of animals (conclusion), indicating that all blue jays are a subset of birds.
What is the hypothesis in the statement 'If two angles are complementary, then they are acute'?
-The hypothesis is 'two angles are complementary'.
What is the conclusion in the statement 'If two angles are complementary, then they are acute'?
-The conclusion is 'they are acute'.
Outlines
📌 Understanding Conditional Statements
This paragraph introduces the concept of conditional statements in the context of mathematics and logic. It explains that a conditional statement is structured as 'if p then q', where 'p' represents the hypothesis and 'q' represents the conclusion. The paragraph uses visual aids like Venn diagrams to help understand the relationship between the hypothesis and conclusion. Examples are given to illustrate how to identify the hypothesis and conclusion in different conditional statements, such as 'if today is Thanksgiving day, then today is Thursday'. The paragraph also discusses variations of the 'if p then q' form, like 'p implies q' and 'p only if q', emphasizing that they convey the same idea but are phrased differently.
📐 Applying Conditional Statements with Examples
The second paragraph delves into applying the concept of conditional statements with practical examples. It demonstrates how to formulate conditional statements by identifying the hypothesis and conclusion in given scenarios. The examples include identifying properties of obtuse triangles, using Venn diagrams to relate blue jays to birds, and discussing the relationship between complementary and acute angles. The paragraph emphasizes the importance of specifying the subject in conditional statements for clarity, such as stating 'if an animal is a blue jay, then it is a bird'. The goal is to practice recognizing the components of a conditional statement and expressing them in the 'if p then q' format.
Mindmap
Keywords
💡Conditional Statement
💡Hypothesis
💡Conclusion
💡Venn Diagram
💡Rational Number
💡Obtuse Triangle
💡Complementary Angles
💡Blue Jay
💡Implication
💡If-then Form
Highlights
Introduction to conditional statements in section 2, 2 notes.
Definition of a conditional statement as 'if p then q'.
Explanation of 'p' as the hypothesis and 'q' as the conclusion.
Visual representation of conditional statements using symbols or a Venn diagram.
Example 1: Hypothesis is 'today is Thanksgiving day', conclusion is 'today is Thursday'.
Example 2: Hypothesis is 'a number is an integer', conclusion is 'a number is a rational number'.
Example 3: Hypothesis is 'a number is divisible by 6', conclusion is 'a number is divisible by 3'.
Alternative ways to write 'if p then q' such as 'if p, q', 'p implies q', and 'p only if q'.
Instruction to write conditional statements in 'if-then' form.
Example A: 'If a triangle is obtuse, then it has exactly one obtuse angle'.
Using a Venn diagram to illustrate the hypothesis and conclusion.
Example B: 'If an animal is a blue jay, then it is a bird'.
Example C: 'If two angles are complementary, then they are acute'.
Emphasis on identifying the hypothesis and conclusion in conditional statements.
Practical exercise in writing conditional statements from given examples.
Importance of specifying the subject in conditional statements.
Summary of the process for identifying and writing conditional statements.
Transcripts
hi everyone we're going to talk about
section 2
2 notes today which are dealing with
conditional statements so let's first
talk about what a conditional statement
is
a conditional statement is a statement
that
can be written in the form if p then
q so you'll see the conditional
statement written out in that form
p then an arrow q well what does the p
stand for and what does the q stand for
the p stands for the hypothesis and this
will follow the statement
when you see the word if the conclusion
is the q part of the conditional
statement
and it follows the word vet okay
we can see it as the symbols or in a
venn diagram
where p in blue is our hypothesis
and then q in red
being the conclusion okay
so let's look through a couple of
examples see if we can pick out what the
hypothesis
and conclusion would be of these
conditional statements
so remember the hypothesis from our
terms up here
will follow the word if so our example
if today is thanksgiving day then today
is thursday so the hypothesis
here following the word if is in purple
today is
thanksgiving
day and then our conclusion
would follow then
which is this part in blue so the
conclusion
today is thursday
so we're just gonna go through and
identify what the hypothesis
and what the conclusion would be let's
try example b
a number is a rational number if
it is an integer so let's look for those
key words
remember the statement that follows if
would be the hypothesis so we look for
that word if
that means this is going to be the
hypothesis
okay now when we write out our part of
the statement here
we don't want to just say it is an
integer we want to refer to well what
is it well in this case it is referring
to a number
so we'll say a number
a number
is an integer
and then our conclusion here so it's
a little bit flipped around
would be this in pink so conclusion
a number is
a rational
number so you want to look for those key
words to kind of help us pick out
the hypothesis and conclusion all right
let's try another one
letter c a number is divisible by three
if it is divisible by six so again this
one's flipped around look for that word
if which means this in pink is the
hypothesis
again we don't want to just say it is
divisible by 6
what is it referring to in this case a
number
so we'll say a number
is divisible
by 6 so that will be our hypothesis
and then our conclusion
will be what's in green so a number
is divisible
by three okay we have a little sentence
here it says
if p then q can be written as
if p comma q q
if p that's kind of like how letter b
and letter c
were p implies q
and p only if q so all of those can be
written
as the if p then q okay they would all
still
be the same type of idea just written a
little bit differently
okay so then we're going to go through
and write a few examples
of some conditional statements so
starting with letter a it says
an obtuse triangle has exactly one
obtuse angle so for us to write this in
if then form we can say if
a triangle
is obtuse so we have our
if p then q then
it has exactly
one obtuse
angle okay so in this case our
hypothesis
would be that it's an obtuse triangle
our conclusion
that it has one obtuse angle so written
if then form if a triangle is obtuse
then it has exactly one obtuse angle
letter b is using a venn diagram so if
we look up at the beginning of our notes
our hypothesis p will be inside the
larger circle
or oval the conclusion q
so we see blue jays would be our
hypothesis
p and birds would be our conclusion
q so for us to write conditional
statement we'll say
if and blue jays are animals so
i'll say if an animal
is a blue jay
then it is
a bird okay and again i wanted to
specify
what i'm referring to instead of just
saying if
it is a blue jay i specify that if an
animal is a blue jay then it is a bird
so again putting that
in that if p then q
statement conditional statement and then
our last example there
part c we have two angles that are
complementary
are acute so our hypothesis
and then our conclusion so let's write
that out as a statement
if two angles
are complementary
complementary then
they are acute and there it would be
in that if-then form so just practicing
being able to identify the hypothesis
what the conclusion is and then write
some of those statements
in the conditional statement format
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