2.2 Notes Part 2

Summary

This lecture focuses on evaluating the truth value of conditional statements in the format 'if p then q'. It explains that a statement is false only when the hypothesis (p) is true and the conclusion (q) is false. Examples are provided to illustrate how to identify true and false statements and give counterexamples. The lecture also covers the concepts of negation, converse, inverse, and contrapositive of a statement, using examples to demonstrate how to write and determine their truth values. The importance of understanding logical equivalence between a conditional statement and its contrapositive, and between the converse and inverse, is emphasized.

Takeaways

• 📌 Conditional statements are evaluated as false only when the hypothesis is true and the conclusion is false.
• 🔍 A single counterexample is sufficient to prove a conditional statement false.
• 🗓️ Example: 'If this month is August, then next month is September' is a true statement.
• 📏 Example: 'If two angles are acute, then they are congruent' is false, as shown by 30 and 45-degree angles.
• 🔢 Example: 'If a number greater than two is prime, then five plus four equals eight' is true because the hypothesis is false (4 is not prime).
• ❌ Negation of a statement (not p) flips the truth value: true becomes false, and false becomes true.
• 🔄 The converse of a statement switches the hypothesis and conclusion.
• 🚫 The inverse of a statement negates both the hypothesis and conclusion.
• 🔄 The contrapositive of a statement is the negation of the converse.
• 🔗 Logically equivalent statements (conditional and contrapositive, converse and inverse) share the same truth value.

Q & A

What is the main focus of the script?

The script focuses on explaining conditional statements, specifically how to determine their truth values, and the relationships between different types of statements such as converse, inverse, and contrapositive.

How is the truth value of a conditional statement determined?

A conditional statement is false only when the hypothesis (p) is true and the conclusion (q) is false. Any other combination of truth values results in a true conditional statement.

What is the significance of finding a counterexample for a conditional statement?

A counterexample is crucial because it provides an instance where the hypothesis is true and the conclusion is false, which is the only scenario where a conditional statement is considered false.

Can you provide an example from the script where a statement is true despite having a false hypothesis and conclusion?

Yes, the script gives the example: 'If a number greater than two is prime, then five plus four equals eight.' Here, the hypothesis (an even number greater than two is prime) is false, and the conclusion (five plus four equals eight) is also false, making the overall statement true.

What is the negation of a statement, and how is it represented?

The negation of a statement is the opposite of the original statement. It is represented by inserting the word 'not' into the statement or using the notation '¬p' where 'p' is the original statement.

How do you find the converse of a conditional statement?

To find the converse of a conditional statement 'if p then q', you flip the hypothesis and conclusion, resulting in 'if q then p'.

What is the inverse of a conditional statement, and how is it formed?

The inverse of a conditional statement involves negating both the hypothesis and conclusion. For 'if p then q', the inverse is 'if not p then not q'.

How is the contrapositive of a conditional statement derived?

The contrapositive is formed by negating both the hypothesis and conclusion and then swapping them. For 'if p then q', the contrapositive is 'if not q then not p'.

Why are the conditional and contrapositive of a statement considered logically equivalent?

The conditional and contrapositive are logically equivalent because they share the same truth value. If the conditional is true, the contrapositive is also true, and vice versa.

What is the relationship between the converse and inverse of a statement?

The converse and inverse of a statement are also logically equivalent, sharing the same truth value. If the converse is true, the inverse is true, and if the converse is false, the inverse is false.

Can you provide an example from the script that illustrates the concept of logically equivalent statements?

Yes, the script uses the example of a cat having four paws. The conditional statement 'If an animal is a cat, then it has four paws' and its contrapositive 'If an animal does not have four paws, then it is not a cat' are both true, making them logically equivalent.

Outlines

00:00

📚 Understanding Conditional Statements

This paragraph discusses the concept of conditional statements in the 'if p then q' format. It emphasizes that a statement is only considered false when the hypothesis (p) is true and the conclusion (q) is false. The instructor guides the audience through examples to determine the truth value of statements, such as 'if this month is August, then next month is September', and explains the concept of counterexamples. The paragraph also introduces the negation of a statement, denoted by 'not p', and its implications on the truth value.

05:00

🔍 Exploring Converse, Inverse, and Contrapositive

The second paragraph delves into the related concepts of converse, inverse, and contrapositive of a conditional statement. It provides a step-by-step guide on how to formulate these statements from a given conditional 'if p then q'. The paragraph uses examples such as 'if an animal is an adult insect, then it has six legs' to illustrate how to derive the converse ('if an animal has six legs, then it is an adult insect'), inverse ('if an animal is not an adult insect, then it does not have six legs'), and contrapositive ('if an animal does not have six legs, then it is not an adult insect').

10:03

📉 Evaluating Truth Values of Statements

In this final paragraph, the focus is on evaluating the truth values of the original conditional statement and its derived forms: converse, inverse, and contrapositive. Using the example of a cat having four paws, the paragraph explains how to determine if these statements are true or false. It highlights the logical equivalence of a conditional statement and its contrapositive, as well as the converse and inverse, meaning they share the same truth value. The instructor concludes by encouraging students to upload their notes and reach out for clarification if needed.

Keywords

💡Conditional Statements

Conditional statements are logical expressions that evaluate to either true or false based on whether the condition specified is met. In the video, conditional statements are discussed in the context of 'if p then q' format, where 'p' is the hypothesis and 'q' is the conclusion. The video emphasizes that such statements are only false when the hypothesis is true and the conclusion is false, which is a fundamental concept in logic and critical thinking.

💡Truth Value

Truth value refers to the truthfulness of a statement, which can be either true or false. The video script discusses determining the truth value of conditional statements by providing examples and counterexamples. It's central to understanding logical reasoning and is used to evaluate the validity of arguments within the script.

💡Hypothesis

A hypothesis is a proposed explanation or assumption made based on limited evidence, which can be tested or verified. In the context of the video, the hypothesis is the 'p' in a conditional statement 'if p then q'. The script uses examples to show that for a conditional statement to be false, the hypothesis must be true while the conclusion is false.

💡Conclusion

A conclusion is a statement that follows and is supported by premises or arguments. In the 'if p then q' format, 'q' represents the conclusion. The video explains that the truth value of a conditional statement depends on the relationship between the hypothesis and the conclusion, with the statement being false only if the hypothesis is true and the conclusion is false.

💡Counterexample

A counterexample is an instance that disproves a claim or a general statement. The script mentions finding a counterexample to show that a conditional statement is false, which involves identifying a scenario where the hypothesis is true but the conclusion is false, thus disproving the statement.

💡Negation

Negation is the act of denying or contradicting a statement. In logic, the negation of a statement 'p' is often written as 'not p'. The video explains the concept of negation in relation to conditional statements, emphasizing that the negation of a true statement is false, and vice versa.

💡Converse

The converse of a conditional statement 'if p then q' is 'if q then p'. It is formed by reversing the hypothesis and the conclusion. The video script uses examples to illustrate how to find the converse of a statement and discusses its truth value in relation to the original conditional.

💡Inverse

The inverse of a conditional statement 'if p then q' is 'if not p then not q'. It involves negating both the hypothesis and the conclusion. The script explains how to derive the inverse from a given conditional statement and discusses its logical relationship to the original statement.

💡Contrapositive

The contrapositive of a conditional statement 'if p then q' is 'if not q then not p'. It is formed by negating both the conclusion and the hypothesis and then reversing them. The video script clarifies that the contrapositive of a statement is always logically equivalent to the original statement, meaning they share the same truth value.

💡Logically Equivalent

Logically equivalent statements are those that have the same truth value in every possible scenario. The video script explains that a conditional statement and its contrapositive are always logically equivalent, as are the converse and inverse of a statement, meaning they either both hold true or both are false in any given context.

💡Acute Angles

Acute angles are angles that are less than 90 degrees. The video uses acute angles as an example to discuss the concept of a false conditional statement. It mentions two acute angles, 30 and 45 degrees, to illustrate a case where the hypothesis is true but the conclusion (that they are congruent) is false, making the overall statement false.

Highlights

Conditional statements are only false when the hypothesis is true and the conclusion is false.

A counterexample is needed to show that a conditional statement is false.

If this month is August, then next month is September is a true statement.

If two angles are acute, they are not necessarily congruent.

An even number greater than two is not prime, making the statement true.

The hypothesis being false makes the overall conditional statement true.

The negation of a statement is written as not p.

The negation of a true statement is false, and vice versa.

The converse of a statement flips the hypothesis and conclusion.

The inverse of a statement involves the negation of both the hypothesis and conclusion.

The contrapositive uses the negation of both the hypothesis and conclusion.

If an animal is an adult insect, then it has six legs is a true conditional statement.

The converse of the insect statement is if an animal has six legs, then it is an adult insect, which is false.

The inverse of the insect statement is if an animal is not an adult insect, then it does not have six legs, which is false.

The contrapositive of the insect statement is if an animal does not have six legs, then it is not an adult insect, which is true.

A cat having four paws is a true conditional statement.

The converse of the cat statement is if an animal has four paws, then it is a cat, which is false.

The inverse of the cat statement is if an animal is not a cat, then it does not have four paws, which is false.

The contrapositive of the cat statement is if an animal does not have four paws, then it is not a cat, which is true.

Conditional and contrapositive statements are logically equivalent.

Converse and inverse statements are also logically equivalent.

Transcripts

all right everybody so we're going to

continue on with our notes so we were

just talking about

conditional statements in that if p then

q

format so now we're going to determine

the truth value

of those conditional statements being

either true or false

okay and the one thing we want to

remember

for determining the truth value here is

that the statement

is only false when the hypothesis

is true and the conclusion is false

so it says to show that conditional

statement is false

you need to find only one counter

example so we talked about those in the

last section

where the hypothesis is true and the

conclusion is false

so we're going to read through some of

these examples and figure out

are these statements true and if there's

if they're not true

we'll give a counter example so it says

if this month is

august then next month is september

we want to think about would that

statement would be true

if this month is august then next month

is september that would be

a true statement when we think about it

letter b if two angles are acute

then they are congruent well let's think

of two acute angles let's say we have

an angle being 30 degrees and an

angle being 45 degrees both of these

would be acute angles

and they're not congruent they don't

have the same measure

so this would be a false

statement okay let letter c

if a number greater than two is

prime then five plus four is equal to

eight so let's

break this down if an even number

greater than two is prime well if we

think about our prime

numbers again with the factors of one

and the factor itself

so our prime numbers if i list out some

of these

well if i think about an even number

greater than

twos so an even number will say four

okay four is not a prime number so this

part right here is false

okay this part right here

five plus four equals eight that is

false also

okay well this overall statement then

would be true now let's talk about that

that seems a little odd well if we look

back up here

our statement it is only false when the

hypothesis is

true and the conclusion is false okay

since our hypothesis is false

and our conclusion is false this is an

overall true statement

we know that 4 is not going to be prime

well then 5 plus 4 does not equal 8

either

okay so we just want to be careful of

those and remember that

the statement is only going to be false

when the hypothesis

is true and the conclusion is

false there okay so it says if the

hypothesis is false so this is referring

to

part c that we just did our hypothesis

was false

the conditional statement is true

regardless of the truth value of the

conclusion

okay so once we saw that

an even number greater than 2 as prime

was false

it doesn't matter about the conclusion

the statement will be

true overall okay the next part of our

notes talks about the negation of a

statement

p and writing that as not p

and you'll see that the notation for

this

is that little tilde and then p

okay so whenever we see the negation

here

we're going to insert the word not into

our statement

so the negation of a true statement is

false

and the negation of a false statement is

true

so again we'll kind of practice with

these in some of the next examples

so we've talked about what the

conditional statement was in section two

one

if p then q okay we had our hypothesis

being p

then our arrow and then q the conclusion

well we also have what is called the

converse of the statement

in order to find the converse you'll see

our symbols right here

we'll flip around the hypothesis and the

conclusion

so then we'll say if the conclusion then

p the hypothesis so we'll flip those two

things

the next type of statement is the

inverse so this

is going to involve the negation okay

and you'll see those little little

symbols there

not p then not q so again any time you

see

that tilde that means not

okay or the little squiggly line and

then the contrapositive

again we'll use the tilde then negation

not q and then not p

okay so we're gonna practice with just

writing some of these statements

being able to figure out what's the

converse what's the inverse and what's

the contrapositive

so let's start with part a if an animal

is an adult insect then it has six licks

so let's just identify well what is the

hypothesis what is the conclusion

this is our conditional statement if p

then q

so o is an adult insect that's p

and then it has six lays legs that's q

okay so that's just picking out what our

hypothesis

and conclusion is so then if we want the

converse

we want to write the converse of that

statement if q

then p so then we're going to flip this

statement around

so i'll say if

an animal

has six legs

then it is

an adult insect

okay so that would be our statement

written as the converse here

insect okay

then let's go through and write the

inverse so the inverse

not p then not q so

all we're going to do is take the

conditional statement which was this one

right here

and put the word not in the hypothesis

and conclusion so going back

up here if an animal

is not an adult

insect

then it

does not

have six legs

okay and then the last statement we're

going to write is

the contrapositive which was this one

right here

not q then not p so we can actually go

back to the converse which was this one

which was q then p and then just insert

the word

not into that statement so if

an animal

does not have six legs

then it is not

an adult insect

okay so for this example we just wanted

to go through and write the statements

okay we'll go through an example b and

find the truth values after we've

actually gone through and written the

sequence okay so

based on the conditional c statement if

p then q

then you can come up with the converse

the inverse

and the contrapositive all based on that

conditional

okay so let's try a letter b

if an animal is a cat then it has four

paws so here's our conditional statement

if p

then q so if an animal is a cat

hypothesis

then it has for pause q so our converse

will flip that around q then p

okay so we would write if

an animal again i don't want to say just

if it has four paws if an animal has

four paws

then it is

a cat which is the hypothesis

all right let's write the inverse the

inverse not

p then not q so we just have to go back

to our conditional and insert the word

not

so if an animal

is not a cat

then it

does not

have four

paws okay there's our inverse

our contrapositive will be not

not q then not

p so let's go to our converse which was

q then p and

insert the word not so if

an animal

does not have four

paws

then it

is not a cat

okay so now let's go back and look at

our truth values so when we

read through these statements would they

be true or would they be false so let's

start with

our original conditionals statement if

an animal is a cat

then it has four paws okay this would be

true again we're thinking of the normal

case where all cats have four paws

okay the converse if an animal has four

paws

then it is a cat well this will be false

because

a dog has four paws

right so this would have this would be a

false statement

okay if it has for a pause doesn't

necessarily mean it is a cat

the inverse if an animal is not a cat

then it does not have four paws

well if it isn't a cat let's say it is a

dog

okay

um that doesn't necessarily mean that it

doesn't have four

paws okay and the contrapositive

if an animal does not have four paws

then it is not a cat this statement

would be true if you read through that

so when we think about conditional

statements

okay if we look at our example with the

cat here

the statements that have the same truth

value are called logically equivalent

statement

a conditional and is contrapositive are

logically equivalent

and so are the converse and inverse so

what does that mean

well if we look at our cat example the

conditional

and the contrapositive were both true

so they have the same truth value there

the converse and the inverse were both

false

okay so they'll follow each other let's

say that

the conditional and the contrapositive

or both false

and vice versa the converse and inverse

being both

true so again they're going to be

logically equivalent they will have the

same truth value there

okay all right so that ends our section

two two notes so you can go

ahead and upload those to canvas again

if you have any questions on this

section

please feel free to reach out to your

teacher so you can

get some explanations have a wonderful

day