2.2 Notes Part 1

Summary

This 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

00:00

📌 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.

05:02

📐 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.

Keywords

💡Conditional Statement

A conditional statement is a fundamental concept in logic and programming that describes a scenario where the execution of a certain part of code or a logical conclusion depends on whether a particular condition is true. In the video, conditional statements are introduced with the form 'if p then q', where 'p' is the condition or hypothesis and 'q' is the conclusion that follows if 'p' is true. The video uses this concept to explain logical constructs and to illustrate how certain outcomes depend on specific conditions being met.

💡Hypothesis

In the context of the video, a hypothesis is the condition or statement that precedes the word 'if' in a conditional statement. It represents a premise that must be true for the conclusion to follow. For example, in the statement 'If today is Thanksgiving Day, then today is Thursday,' 'today is Thanksgiving Day' is the hypothesis. The video emphasizes identifying the hypothesis as a key step in understanding conditional logic.

💡Conclusion

The conclusion in a conditional statement is what follows the word 'then' and is the outcome or result that occurs if the hypothesis is true. It represents the logical consequence of the hypothesis being valid. In the script, 'today is Thursday' is the conclusion that follows if the hypothesis 'today is Thanksgiving Day' is true, illustrating the direct relationship between hypothesis and conclusion.

💡Venn Diagram

A Venn Diagram is a graphical tool used in the video to visually represent the relationship between a hypothesis and a conclusion in a conditional statement. It uses overlapping circles to show the logical inclusion of one set (hypothesis) within another (conclusion). For instance, the video describes a Venn Diagram where 'p in blue' (hypothesis) is contained within 'q in red' (conclusion), helping viewers to understand the logical structure of conditional statements.

💡Rational Number

A rational number is a number that can be expressed as the quotient or fraction of two integers. In the video, rational numbers are used as an example in a conditional statement: 'A number is a rational number if it is an integer.' This example helps to clarify how a specific type of number (integer) falls under a broader category (rational number) based on a conditional relationship.

💡Obtuse Triangle

An obtuse triangle is a type of triangle that contains an angle greater than 90 degrees. The video uses this concept in a conditional statement: 'If a triangle is obtuse, then it has exactly one obtuse angle.' This statement helps to understand that the presence of an obtuse angle is both necessary and sufficient to define an obtuse triangle.

💡Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. The video mentions this concept in the context of a conditional statement: 'If two angles are complementary, then they are acute.' This statement illustrates the logical relationship where the sum of angles being 90 degrees implies that each angle must be less than 90 degrees (acute).

💡Blue Jay

A blue jay is a specific type of bird used as an example in the video to demonstrate the use of a Venn Diagram and conditional statements. The script states, 'If an animal is a blue jay, then it is a bird.' This example shows how a specific instance (blue jay) can be logically included within a broader category (birds), reinforcing the concept of conditional statements.

💡Implication

Implication in logic is the relationship between a hypothesis and a conclusion where if the hypothesis is true, then the conclusion must also be true. The video mentions 'p implies q' as a way to express a conditional statement, emphasizing the causal link between the hypothesis and the conclusion. This is crucial for understanding how logical conclusions are drawn from given premises.

💡If-then Form

The if-then form is a common way to express conditional statements, as introduced in the video. It is a clear and concise way of stating that if a certain condition (if) is met, then a specific outcome (then) will occur. The video provides several examples of how to write statements in if-then form, such as 'If a triangle is obtuse, then it has exactly one obtuse angle,' which helps in constructing logical arguments and programming conditions.

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