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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

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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

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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

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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

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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

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[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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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