2.2 Notes Part 2
Summary
TLDRThis 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
- π Conditional statements are evaluated as false only when the hypothesis is true and the conclusion is false.
- π A single counterexample is sufficient to prove a conditional statement false.
- ποΈ Example: 'If this month is August, then next month is September' is a true statement.
- π Example: 'If two angles are acute, then they are congruent' is false, as shown by 30 and 45-degree angles.
- π’ 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).
- β Negation of a statement (not p) flips the truth value: true becomes false, and false becomes true.
- π The converse of a statement switches the hypothesis and conclusion.
- π« The inverse of a statement negates both the hypothesis and conclusion.
- π The contrapositive of a statement is the negation of the converse.
- π 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 'Β¬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
π 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.
π 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').
π 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.
Mindmap
Keywords
π‘Conditional Statements
π‘Truth Value
π‘Hypothesis
π‘Conclusion
π‘Counterexample
π‘Negation
π‘Converse
π‘Inverse
π‘Contrapositive
π‘Logically Equivalent
π‘Acute Angles
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
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