Understanding the Contrapositive, Converse, and Inverse

Video Series: Logic and Mathematical Proof (In-Depth Tutorials for Beginners)

(D4M) — Here is the video transcript for this video.

00:00
in this video we delve into understanding the contrapositive
converse and inverse of an implication the contrapositive of a statement can be
a powerful tool for proving mathematical theorems
because the contrapositive of a statement always has the same truth
value as the statement itself and especially if the truth of the
contrapositive is more accessible to establish than the truth of the original
statement so let’s see what they are hi everyone welcome back i’m dave
this video understanding the contrapositive converse and inverse
is a video in the video in the series logic and mathematical proof
in-depth tutorials for beginners so let’s go ahead and get started
what is the contrapositive converse and inverse of an implication
and then i’m going to explain how they’re related to each other
and then we’re going to write the converse contrapositive inverse

00:01
and practice using some examples there so let’s get started what is the
contrapositive converse um of a implication right and and the
inverse and so we’re going to see that right now
so three important statements are associated with any implication p implies q
so that’s the important thing is to realize is that we have an implication
and only then can you talk about contrapositive converse and inverse
so an implication is given to us so for example let’s say we’re given um
1 plus 2 equals 3 implies 1 plus 3 equals 5.
so here we have an implication let’s call this our p and this our q
and so we have p implies q so that’s given to us so now we can ask

00:02
what is the contrapositive of this so the contrapositive would be not q
implies not p and we can write that out in terms of
the original statements so what is not qs right so what’s the naught of q
so one plus three is not equal to five implies not q one plus two
is not equal to three so this is our original implication
and this is our contrapositive right here this is the contrapositive right here
now we can also write the converse of p implies q
so this would be the contrapositive and the converse would be q implies p
so what would the converse be so it’d be 1 plus 3 equals 5 that’s the q
implies p which is 1 plus 2 equals 3. so this would be the converse right here

00:03
and then finally what would be the inverse so the inverse would be not p
so 1 plus 2 equals 3 is not equal to 3 that’s the not p implies the not q
so 1 plus 3 is not equal to 5. so this is the inverse right here
so we have the inverse we have the converse we have the contrapositive and
we have the implication so all four of those
is something that you can talk about whenever you have an implication you can
talk about the implication itself or you can talk about these related statements
now how are all these statements related to each other and so let’s see that now
so how are they all related to each other so we’re going to show that the
contrapositive and the original statement are
equivalent to each other and we’re going to show that the converse and the

00:04
inverse all right so let’s see that now so here we go show that the implication
and the converse are equivalent to each other
so here is the implication right here so let’s go with some p here
and then we got some q so let’s go true false true false true true false false
by the way i’d like to say this moment that this video here um
is part of a series and we already talked about how to
you know what these logical connectives are we’ve already talked about how to
construct truth tables and we’ve already talked about tautologies and
contradictions and stuff like that so let’s do the implication right here

00:05
[Music] so the implication is true false true true and now let’s look at the um
[Music] we also need the not p and not q so let’s look at not p and not q right
here so this will be false false true true and this will be false true
false true and now let’s look at the um let’s look at say the contrapositive
next not q implies not p so here’s the contrapositive here
so we’re looking at not q implies not p so false implies a false is true
a true implies a false is false a false implies a true is true
the true implies the false is true so you can see these two columns right here
have the exact same values right here so these are equivalent to each other

00:06
so we can say not q implies not p well we ran out of room there so we can say
this statement right here q1 and q2 so we can say q1 is equivalent to q2
right because true and true is true false equivalent to false is true
these match right here so true and these match right here so true again
so this is tautology right here so the state of the implication and the
contrapositive are always equivalent to each other all right so
that takes care of number one now for number two notice that the
inverse and converse are logically equivalent so let’s look at that now
so let’s come over here and do the inverse and the converse so here we need

00:07
p and q so true false true false true true false false
and we’re also going to need the um negations so not p and not q
so not p would be false false true true and not q would be false true false true
and then now let’s look at the converse q implies p
that’s the converse right there what is q implies p so we’re looking at q
implies p so looking at true implies a true false implies a true
implies the false also implies a true so that’s the converse right there
and now let’s look at the inverse so the inverse is not p implies not q

00:08
so we’re looking at the implication right here not p implies not q
so false implies a false is true false implies a true is true
true implies a false is false and then a true implies a true is true so
we got this uh two things right here let’s call this the q1 and
q2 and so the last column right here will be q1 if and only if q2
so you see they match and they match so this right here
so this right here says the the um converse and the inverse
match they’re logically equivalent in other words the equivalence is the
tautology so the that takes care of number two

00:09
notice the inverse and converse are logically equivalent all right so um
now the next thing would be to look at is an implication is not necessarily
equivalent to its converse so here’s an implication right here
and where’s the converse right here here’s the converse right here in this
column right here so we have true true false true and here we have an
implication true false true true so the the the implication and the
converse are not uh sorry number three is inverse and
converse so the converse is right here so i need to be looking right here and
right here at these two true true false true and this is true
false true true so the implication and it’s converse are not equivalent to each
other and then the same thing with number four the implication is not
necessarily going to be equivalent to its inverse
right here so you can see this is the converse this is the inverse right here

00:10
and while these two match they don’t they don’t have the same
value as the implication here so they’re not necessarily equivalent
even though the converse and inverse are equivalent and the implication is
equivalent to its contrapositive so these are equivalent to each other
and these are equivalent to each other but these are not equivalent to each
other right here okay so there’s the relationship between those four
and now let’s think about um the um you know let’s write out some examples now
so let’s write the converse contrapositive and inverse
by looking at these examples right here so let’s do this now
so let’s look at number one here first so we have number one here

00:11
and it’s saying we’re going to say p is x is not equal to 0
and we’re going to say q is x squared is not 0
and we have an implication p implies a q right so that’s what we’re given
and we want to write the contrapositive right so the contrapositive is not q
implies not p and we want to write the converse and we want to write the inverse
so contrapositive converse and inverse so what would the contrapositive be we
would do not q so what’s the naught of a negat
of a negation already so this would be if x squared is equal to zero

00:12
then and then now we need to do the not p so the not p would be equal zero
so this would be the contrapositive here if x squared is equal to zero then x is
zero now in this problem here we’re not trying to determine if things are true
or false we’re just simply writing an implication and writing out different
forms of that implication this this is the contrapositive right here
and this right here is the converse so q implies not p so this would be the
statement if q so if x squared is not equal to zero
then and then now the p which is x is not zero
right so this is the converse statement right here and this is the inverse so
if not p so if and then negation of p so if it’s x equals zero then
now the negation of q which means x squared is equal to zero

00:13
okay so there’s number one we wrote the um contrapositive um
contrapositive converse and inverse of the implication there given in number one
so now we can just um keep doing this here let’s go on to number two
so number two here we have the same thing so we have p
which is going to be x y equals zero we have q which is x equals zero
and we have an if then we have a p implies a q
and so let’s again write out the um contrapositive
so the not q implies the not p and let’s also write out the converse q implies
p and let’s write out the inverse not p implies not q
so let’s write out each one of these statements here so not q so if

00:14
we’re going to not the q so if x is not zero
then and now we’re going to negate the p x x times y equals is not equal to zero
that’s the contrapositive now we have q implies p so we have q implies a p so if
x equals zero then p which is x times y equals zero
okay now the last one we’re going to be negating the p
so if x times y is not zero then now we need need to negate the q
so negating the q is x is not zero okay so there we go we have the
contrapositive converse and inverse for number two
all right now let’s look at number three so number three here all right so

00:15
we have the p is x is a real number so i’m going to write it in this
notation here x is an element of the real numbers and so then it says then
x is a rational number so that’s my q x is an rational number
so i wrote it in symbols and over here it’s written in english
and we have an implication p implies q that’s the overall structure number three
if p then q so now let’s write the contrapositive
the negation of q implies the negation of p
there’s the contrapositive and now let’s write the in converse
and then let’s write the inverse not p implies not q
so let’s write each of these statements down so the contra positive is
i wanted to negate the q so i’m going to say
if and then we can write it in symbols x is not an element of q

00:16
but since they wrote it in words over here let’s just go ahead and write it
out if x if x no actually i want to write it in symbols not an element of q then
and now we need to negate p x is not an element of the reals
all right that saved me a little space by writing it in symbols now we have q
implies p so q implies p so if x is a rational number then x is a real number
and now we have negation of p so we need
to negate that so if x is not an element of the real numbers then
and we need to negate the q then x is not an element of the rationals
all right so there’s number three let’s try to do number four here number four

00:17
if x is positive so that’s why p x is positive and
l oh we have an and all right so if x is positive and x squared equals four
okay so i have a and here it’s this one and this one and then we have a q is um
x equals two so there’s our q then x equals 2.
so we have an implication p implies q my and is compound in this in this
example here and the q is just a simple simple statement there
then x is equals to two okay so let’s write the contrapositive
not q implies not p and let’s write the converse q implies p
and let’s write the inverse not p implies not q

00:18
let’s write each one of those out so here we go if and we’re going to
negate the q so if x is not equal to 2 then
and now we have the negation of the p so we want to negate this whole thing here
then it’s not true that x is positive and x squared is equal to 4.
now when i want to negate a p and q or let’s say here since i already have a
p here let’s call this q and r so when i want to negate a q and r
i want to use what i want to negate an and so i’m going to put a q and r here
so before i write this out here i want to do a little thinking here so this is

00:19
going to be equivalent to not q or not r okay so i want to use this right here
and this is called the morgans and we’re going to
prove this here in an upcoming video the next video i believe and um
but if you’ve seen the previous video then you should be able to do this just
make a truth table for this statement here oops that’s a q and that’s an r
that’s a q and that’s an r uh make a true table for this
and you’ll see that it’s a tautology so we’re going to actually cover this in
two more videos coming up but we’re definitely covered on the
logical equivalence video but any case when i want to write the
contrapositive of this so i want to negate the q
so i negate the q and then i want to negate the p
but p has got an and in it and so how do we negate an and

00:20
we change the negation of an n to an or and we negate each one of these so then
i would say then x is not positive or x squared is not
equal to four so let’s write that down then x is not positive or
x squared is not equal to four so that would be the statement there if
x is not equal to 2 then x is not positive or x is not equal to 4. so i use the
negation of q or the negation of r and there’s my q and there’s my r
so that would be the contrapositive right there so here’s the original statement
right here if uh if x is positive and x squared equals four
then x equals two and the contrapositive is if x is not two then
x is not positive or x squared is not equal to four

00:21
now let’s write the converse so the converse is the statement just q implies p
so if here’s my q so if x is equal to 2 then so let’s get rid of this space here
then now we need the p so x is positive so if x equals 2 then x is positive and
x squared equals 4. so this would be the q implies p here
if x is equal to two then x is positive and x squared equals four
and that says positive case you can’t read that

00:22
all right so the last one here would be the um the inverse so not p
implies not q so not p implies not q this will be the
inverse here so we want to do the if not p so if now we need the negation of p
now we did the negation of p a minute ago
um so we’re going to say the same thing right here if x is not positive
or x squared is not equal to 4 comma let me let me move up here
so let’s move up there and move the marker board there then
all right so here here we’re doing the inverse not q implies not p
so if and then there’s the nod of p comma then now we need the negation of the q

00:23
so if so then x is not equal to two all right very good so there’s number
four right there all right so now let’s move on to number
five number five is very similar because we also have another and in the
hypothesis there so let’s go on to number five now and i just want to
you know try to reiterate what i’m using here so if we have an and
so say we have a p and a q and i’m doing the negation of an and
and that’s going to be logically equivalent to not p or not q
in fact since we’re going to use it again in number five
let’s go ahead and make sure that this is something something i
called earlier de morgan’s law let’s just make sure that this is actually
um you know a tautology here so let’s actually do that real quick

00:24
before we before we solve number five here so let’s just make sure that what
we’re using right here is is good okay so let’s do that we got
p’s and q’s here so let’s go down here let’s say right here p q
and then we have a not p and a not q and now let’s build the and
and then let’s build the negation so we have the and and the negation and
then let’s build this or right here so not p or not q so let’s go true false
true false true true false false and then false false true true
and then false true false true okay so there we go we got all the
possibilities between p and q and we have the negation of p and the
negation of q now let’s do the and so i’m looking at the p and q here and i

00:25
got true false false false and now let’s negate the end and so i got false true
true true now let’s look at the or between the negations
so here’s the negations right here and let’s look at the or
false and false is false so here we have a true true and true
and so what you can see is that these are the same
anytime you have any true or false from this
exactly matches the true or fall from this they’re both falses trues trues and
trues so you know this right here is is a tautology right here so
i’m going to use this again when i’m working out number five here
just like i did when i worked out number four and again this is called de
morgan’s law to give it a name and let’s see here how we can do this

00:26
so this will be number five right here let me move myself to the bottom here
so let’s go number five here all right so now we have p which is
going to be the x is not equal to 2 and x squared plus 3x plus 1 equals 0.
notice that we’re not solving any equations or anything like that because
we’re just doing logic in this in this video here so here’s the p
and then we have the q this is comes after the then x equals
one and so you can see we have a p implies a q if p then q
this is the p and that’s the q all right so now we’re looking at the
contrapositive so then not q implies the not p
and what will this statement right here be so we need need to negate the q so if

00:27
x is not equal to one then now we need to negate the p we have an and
so i’m going to change this n to an or and negate each one of these
so then x equals two or now negate this one x square plus three x plus one
is not equal to zero so this would be the contrapositive
of number five right here if x is not equal to one then x equals two or
this quadratic is not zero okay so now let’s look at the converse q implies a p
q implies p so i just need to take this q and this p and let’s just write it out
so if x equals one then then the p so then x is not equal to two and
three uh x squared plus three x plus one is equal to zero

00:28
okay so there is the converse right there and now let’s write the inverse
so not p implies not q okay so here we go so if we need to negate the p
so here’s the p up here so i need to negate that so if x is equal to two or
x squared plus three x plus one is not zero then
and then now i need to negate the q so i need to negate that so then so then
x is not equal to 1. not equal to 1. all right and so there’s the inverse
there’s the converse there’s the contrapositive
all right and so now we have number six our last one let’s give that a shot and
see how it goes so let’s look at six here if n is an

00:29
even integer or a prime number then okay and so it looks like we have uh p
implies a q because we have an if then but we also have an or
so now we’re going to actually also use the morgan so what we used here before
was p and q and we made a true table for this to see it
was equivalent to not p or not q so this is what we used on four and five but
there’s actually another version of de morgan’s law which allows us to negate
an or so the negation of an or is no surprise the negation of p and
the negation of q so these are dual statements here you
can think of these as dual statements to negot to sorry to negate an and i
change it to an or of the negations to negate an or i change it to the and

00:30
of the negations now we should see that this is true
we can do that really quick by making a true table if this is the first time
you’ve seen de morgan’s law you should make a truth table to verify this just
like we did for the and make a true table and you’ll see that
this is a tautology i’ll leave that for you to check
and so now let’s go on to solve number six here so we’re not going to use that
one on number six we’re going to use this right here so here we go number six
number six i’m going to say my p is so p comes the after the if before the then
so n is an even integer p is an even integer or a prime number prime number

00:31
and then what’s my q so my q is n is not n is not an even integer
and even non-negative is not an even non-negative integer
okay so there’s the uh p and q that i need now let’s write the uh contrapositive
so the not q implies the not p so this will be our contrapositive here
so we need to negate the q now it already says q is not so negating q is just
in is so here we go we’re going to say if n is an even integer

00:32
an even non-negative integer then so we negated our q so if then and now not p
now when i write the not p i’m going to use this up here so when i’m nodding an
or so because i’m trying to not p and p has an or in it so i’m trying to
negate p now negating an or is i’m going to change it to an and and
then i’m going to negate this one and this one right here so then
n is not an even integer then n is negating this as n is not it is not
an even integer and then now we’re negating an or so it becomes an and and
now the negation of the second one or a prime number or is not a prime number

00:33
so here would be the contrapositive right here
n is an even non-negative integer if then n is not an even integer
what’s that say that’s my or or oh wait no i’m changing or to an and
sorry that said and couldn’t read my own writing and and oh that’s why so and n
is not a prime number okay so there’s the contrapositive there
so now let’s write the um converse right here
um let’s go up here and write it here so here’s for the converse
q and pi is the p so if q and we don’t need to change q at all so if n is not

00:34
an even integer or sorry non-negative integer then
if n is not an even non-negative integer then now we need the p then n is
an even integer now i’m just taking the p so i’m just
writing p verbatim it is an even integer or a prime number
okay so there’s the converse right there
if n is not an even non-negative integer then n is an even integer or a prime
number and so now last but not least let’s look at the inverse so the inverse is
not p implies not q so here’s the inverse right here
so we’re going to negate both of these right here and we need to start with not

00:35
p so not this first so if n now remember we’re negating an or
so i need to negate this right here so if n is not and even integer and in
is not a prime number then and now we need to negate the q
so q doesn’t have any connectives in it to negate q we just change it to an is n
is an even non-negative integer so here we can see the whole statement here

00:36
non-negative integer there we go there’s the inverse
okay so that’s it and if you have any questions or ideas please use the
comment section below uh and don’t forget to look up the
series logic and mathematical proof in-depth tutorials for beginners the
link below the link is below in the description and i want to say thank you
for watching and i’ll see you next time if you like this video please press this
button and subscribe to my channel now i want to turn it over to you math can be
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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