BAM! Mr Tarrou
In this video we are going to introduce using deductive reasoning.
And we're going to be going that by introducing the structure of IF-THEN statements
and then manipulating them through, uh, finding converse statements,
by conditonal statements and showing you how that related to mathematical definitions
that were given in our textbooks
uh, inverse statements and contrapositive statements
and lots of, uh, 'vocal' [vocabulary] in this video.
A few, uh, examples of course in language of how to set-up these IF-THEN statements
but I'm going to be, you know, running though all this, uh,
definition and structure of these statements for you
so you can use them to do your homework
now IF-THEN statements are how we logically go about a complicated problem and we move from,
one given mathematial statement
and, you know, develop that into some new ideas, some new mathematical statement
uh, using steps in the middle that we know, that are work
maybe that are given to us
like an old, maybe an additive addition property from algabra
were you add a number to both sides of an equation
and it still would be true
that's how you move from one line of your work to another line of your work
when you learnt how to solve equations last year in algabra
when we do the two column proofs that are coing up soon your not going to see me writing,
IF-THEN, uh, all over our two column proofs
but that logic that is behind those two column proofs is there
that idea that IF-THEN is behind those, uh, statements
I remember learning some basic and Pascoe programming background was way
way back when I was in school, IF-THEN statements were in that computer
programming language constantly because you know you the computer would be
looking for if the user of the computer types in this then do this command so
there's many many applications
well anywhere that logistics and deductive reasoning can be found you're
going to have this idea if then statements tied-in. So what are they all
here if I save money then I'll be able to retire comfortably if I save money
i'll build
the nest egg and i will have a cushion on of money to use in my retirement I
wouldn't have to work you know until not able to if two lines to do something
mathematical if two lines intersect then they intersect in exactly one point we
proved this theorem it's the first thing we have geometry in the last video if
two lines intersect in the intersection exactly one point well i'm getting into the
structure of the event statement and this yellow here if two lines intercept
the two lines must intercept, for even, for me to even consider what
comes after the word then, ok, to even worry about how many points those two
lines intersected and when you are looking at an IF-THEN statement its if P
then Q it's if and then whatever's after the word if is your condition it is your
hypothesis and it must be true to move on to the rest of the sentence so if P
is true if indeed we do have two lines intersect because they can also be
parallel or skewer they're not the same plane two lines don't have to intersect
but if they are intersecting in that condition is true that hypothesis is
true then Q then they can be on the conclusion therefore you know well must
be true if the IF-THEN statements to be true to approve and IF-THEN statement
false you only need to have one counter-example it's actually very very
difficult to prove a mathematical statement true if you say something in
math if we say well if we say the two lines intersect there is only the only
intersect in exactly one point this must work for all settings, if this were to
work a billion times and fail once then it would be considered mathematically as
a false statement so to prove something true is what we're going to learn a
little bit of there are two column proofs that are coming up but as you
move up in mathematics
it becomes you realize that they can be very complicated process however to
prove something false you'll need one counterexample know what i mean by that
I don't have this ring on the board but if I said you know a quadrilateral
that's a polygon with four sides a four-sided figure a quadrilateral that
has four equal sides or if a quadrilateral has four equal sides then
it is a square if a quadrilateral the four-sided figure has four equal sides
then it is a square that may be a true statement except i think a rhombus
rhombus is a four-sided figure with four equal sides these hatch marks are
indicated those four sides are equal a rhombus has four equal sides but it's
not a square because the corners the angles are not right angle so that would
be a false statement if a quadrilateral has four equal sides then it is
a square
I just came up with a counterexample to that statement so thus that is false and
I just need that one example and I proven you know that statement false
also if you have learned about Venn diagrams you might have this in your
textbook and looking at this if P then Q if P if you have an event in the circle
identified as event
P then Q this is how we display that in the Venn diagram if event p has occurred
all of that circle is inside of the circle where we have event Q this is if
P then Q that's how the Venn diagram would look for a true IF-THEN statement
the Venn diagram that would not represent that
look something like this where we have this is P and this is Q, if P then Q
well part of the circle that goes around letter P part of that circle is not in
Q so there's a pos- what this says is a Venn diagram is there is a possibility
of P occurring or P being true when Q is not because we're not inside that
circle of Q so just be aware that if your book is using Venn diagrams to help
to teach this lesson one book I have doesn't look I have does all right let's
get the next screen the converse the converse of a conditional statement is
for by switch- changing the wording i am i right here the Congress of the
conditional statement is formed by switching the hypothesis and the
conclusion so we have a conditional statement of if P then Q or your
textbook might be using P with an arrow Q were going to talk about a little bit
different language for this IF-THEN statements here in a second then the
converse is if Q then P you just take the phrase that's after the word if and
after the word then and swap 'em or Q arrow P a converse statement may or may
not be true you need to look at the original combined conditional statement
swap the IF-THEN phrases and read that converse and see if you know logically
it makes sense or you know is it true or false
so here's a very kind of ridiculous example and it will be false and that's
not always the case are conditional statement if it rains then i will take an
umbrella
sometimes I don't care if I have an umbrella not i'll just get wet but you
know if it rains
let's just say every time it rains i'll take an umbrella it seemed that could be
a true statement however the converse is if i take an umbrella that it rains the
weather does not care whether I have an umbrella or not if it needs to rain
it'll rain so if i take an umbrella then it rains now
all i need is to prove that false is to prove it false one time so even if i
take the umbrella
maybe I just start taking it out and it rains every time i take umbrella I think wow I'm
influencing the weather but the first time I take that umbrella and it
does not rain
I've proven the statement false again proof IF-THEN statement false or
anything falls for that matter mathematically all you need is one
counter example so erm that's what i just said okay over here other ways of
writing conditional statements other ways of interpreting this IF-THEN so I
have a little very very simple algebraic expression i didn't solve it but we have
if X minus three equals seven then x equals ten and just a one that's just a one-step
algebraic equation to solve that we would add both sides by three to get the
negative three away from the X and see that ok well negative three plus three is zero and seven
plus three does equal ten so it's true and I can check by simply taking ten plug
it in ten minus three is equal to seven ok
at any rate it's a true statement so we can say if P then Q or you can say P
implies Q remember the if the conditioner hypothesis must be true so
key the condition of hypotheses implies Q, X minus three is seven implies that x is equal
to ten, P only if Q, X minus or equal seven only if x equals ten and finally Q if P
the condition if the conclusion excuse me if the conditional hypothesis
is true so x equals ten if X minus three is equal to seven so some other ways of
writing or wording the IF-THEN statement if conditional and it's
converse are both true so if you do have a mathematical statement or any statement for
that matter and you swap the condition or hypothesis and conclusion
if you swap and you still get a true statement
then what you have is a biconditional statement so if P then Q if that's true
statement and then you go if Q then P you write the converse and it is also
true then what you have is the ability to write a biconditional statement and
that comes in here with our if and only if P if and only if Q will see an
example of this in just a second
when i switch to the next chalkboard the key thing to remember with
biconditional statements is every definition that is in your mathematical
textbook can be rewritten as a biconditional statement there's that
direct relationship between biconditional statements and definitions
let's see one and look at how that see how that looks
all right wake you back up again every definition can be read as the biconditional
statement i just said that our definition we have here is the midpoint
of segment is the point that divides the segment into two congruent
segments remember congruent is you know equal in all ways equal shape equal size
and with segments that means that their lengths are equal a biconditional
statement of this definition would be to sort of you know look for that
definition word and then just put if and then if and only if between that in the
definition so a point is a midpoint if and only if we want to focus in on right
here as i walk out of screen if and only if so a point is a midpoint if it only if
it divides the segment into two congruent segments biconditionals, you know, how do you
take a biconditional statement and write it or convert it into conditional
statements well we only have a biconditional IDR setting if we can
write a conditional statement and it's converse and have them both be true so
we want one
given a biconditional statement and we're being asked to convert it into
not a conditional statement but two conditional statements because the
original and the converse must both be true you're going to look both of
those segments are congruent if and only if they have equal length so if segments
are equal in length then they are congruent and if segments are congruent
then they have equal length so I kind of took this biconditional
statement and kinda just worded it in both directions right if P then Q and the
converse if Q then P so I hope that clarifies for you at least in this one
example how to go from a biconditional statement to your two conditional
statements we're going to look at now on what or how to setup an inverse
statement and a contrapositive statement this video will be done here we have a
conditional statement if the measure of angle A is equal one-hundred-and-ten degrees then
angle A I took away the end the M measurement indicator is obtuse because i'm
talking about certain name in the angle based on its size not measuring how many
degrees that angle opens up by the way not looking at it needs a degree mark
so the inverse you negate sort of you write the expression as a negative
statement negate both hypotheses and the conclusion of the condition you take the
language that is there and you switch it if something is happening that it's not if
something's not happening or not true well then it is true you indicate you
swap it that's what this little squiggly sign here means so if not P then not Q
so if the measure of angle A is not equal to one hundred and ten degrees supposed to be
an equal to one hundred and ten degrees
then this is a stretch by by the way here this is not a six then angle A is
not obtuse
ok let's look at the original conditional statement if the measure of
angle A is 110 degrees then angle A is obtuse, well, obtuse angles have measures that
are greater than 90 less than 180 degrees indeed that is true for 110
degrees our original IF-THEN statement is true our inverse statement like your
converse as they may or may not be true you need to look at the inverse
statement and decide that for yourself and with the setting that is given if an
angle is not a hundred ten degrees then it is not obtuse this can be stated as
being a false statement if you can think of even one angle in this example that
will prove this to be true in other words this is saying that the only this
is kinda basically saying that only way for an angle to be obtuse so it's if it's a
hundred and ten degrees if it's not a hundred and ten then it's not obtuse but i
can think of another angle measure that, or I can think of an angle measure that is
not a hundred and ten and is obtuse how about one hundred and fifteen degrees is not
a hundred and ten but it is an obtuse angle so this is a false statement
okay just because an angle is not a hundred and ten degrees does not mean
that it is or does not mean that it is not obtuse. Contrapositive you want to
negate both the hypothesis and the conclusion of the, and here's
something to pay attention to, of the converse so you're not only going to
negate the expression behind the if and then but you're going to write the
converse statements which is when we take the condition and the conclusion
and swap so we have not Q implies or then not P so we still have the not
symbols up here
but the P and Q have swapped places because we're talking about the converse
statement so if angle A is not obtuse which is what it says here then measure
of angle A is not equal to 110 so before you really quick say it's false
let's make sure because again you need to look at these statements
individually and determine if they are true or if they're false with your
inverse, your contrapositive, your converse statements they may or may not be true
so if angle A is not obtuse well if angle A is not obtuse then it is not
greater than 90 and less than a hundred eighty degrees so it could be a hundred
eighty could be 90 could be 70 could be 60 could be many many different angles
then it is not then the measure of angle A is not a hundred and ten if angle A is
not obtuse then measure is not a hundred and ten all the angles that i just
discussed about it not being obtuse, being 180, being 90, less than 90 down to zero this
is a true statement if the measure or if angle A is not obtuse if it is not greater
than 90 and less than 180 then it is not going to equal 110 degrees
I don't know what the measurements gonna be equal to but I know it's not gonna be a
hundred and ten because a hundred and ten is an obtuse angle so
this is true and this is your induction for me anyway of using deductive
reasoning and your introduction to the IF-THEN statement my next video is going
to be introducing the structure of a two column proof so I'm Mr Tarrou, BAM!
go do your homework!