In this video, I will use examples
to explain the various pieces of notation we use in mathematics
to talk about conditional statements, their precise meaning,
and what conclusions we can or cannot draw
when we find a conditional.
A conditional statement is something that has the word "if."
And in mathematics, we often write it as "if P, then Q,"
where P and Q are statements themselves.
Sometimes, we abbreviate it with this symbol, this double arrow.
And this double arrow is read, "implies."
So these three lines, they all mean the same thing;
they are entirely equivalent.
What does a conditional statement mean?
When I have "if P, then Q,"
that means that whenever P is true, Q must be true as well.
And that's it.
That is ALL it means, nothing more.
So to be very precise I'm going to spell out that in particular,
we're saying NOTHING whenever P is false; whenever P is false, we don't care.
So let me write a very basic example to show
that this is ACTUALLY what this means, and nothing more.
This is a statement about real numbers, and I'm saying
that IF x is greater than 10, THEN x must be greater than 6.
And I hope you agree; this is clearly true.
But let's look at a few specific values of x.
Let's look at x = 12.
In this case, the "IF" part is true, and the "THEN" part is also true.
When x = 8, x is NOT greater than 10; the "IF" part is false,
but the "THEN" part is still true.
When x = 3, they're BOTH false.
Okay. So this example illustrates all the possible cases.
As I said earlier, "if P, then Q" means
that whenever P is true, Q must be true as well.
And whenever P is false, we don't care; it can be anything.
To test that that is clear, here is a little exercise for you.
I have a subset of the real numbers which I am calling A,
and assume we know it is true for this set
that if x is an element of A, then x must be positive.
What can I conclude in each one of these 3 cases
about a real number x?
Pause the video for just 15 seconds, think about it,
and then continue watching.
So we know that if x is in A, then x must be positive.
If x is not in A, can I conclude something about x?
Not at all; it's exactly what we said before.
The "IF" part is false, so I know nothing else.
What about if x is positive? Can I conclude something about x?
Not really.
PERHAPS x is positive because x is in A,
but perhaps x is not in A and then x is still allowed to be positive.
So in the second case also, we have no conclusion.
But what if x is NOT positive?
Ah! In this case, I CAN conclude that x is NOT an element of A.
And why is that?
Because if x were an element of A, it would be forced to be positive.
So in order to be less than or equal to 0,
it must be OUTSIDE of A.
And this is actually a very general result:
Whenever you have an implication like "P implies Q,"
you get, for free, another implication which is "not Q implies not P".
"P implies Q" and "not Q implies not P,"
this one and this one, mean EXACTLY the same thing.
Be careful: when you have "P implies Q,"
you CANNOT conclude that not P implies not Q.
But you can ALWAYS conclude that not Q implies not P.
This comes in handy.
Great! We now understand the basic notation for conditionals.
I need to introduce
a second piece of notation that is related, but different.
Let's look at this new example. Now I'm looking at integer numbers.
Let's compare these two statements: n is even, and n is a multiple of 4.
If n is even, can I conclude that n is a multiple of 4?
No, for example n = 6 is even, but it's not a multiple of 4.
On the other hand,
if n is a multiple of 4, THEN I can conclude n is even.
So IF this, THEN this.
And I can write it
with the implication arrow going backwards.
The pointy end towards the conclusion.
Next, let's compare these two statements: n is even and n + 1 is odd.
If n is even, then certainly, n + 1 is odd.
But now, I can also go backwards;
if n + 1 is odd, then n must be even as well.
So I can say both implications so why not write them together?
And the standard notation
is to use the double arrow, but make BOTH ends pointy.
This symbol here is called "if and only if,"
and when written, it is often abbreviated
as if with a double F ("iff"). But spoken, it's always "if and only if."
And it matters that we are able
to distinguish between a plain "if," and an "if and only if."
So, to spell it out, when I write "P if and only if Q,"
what I'm actually saying is that two things are true:
"if P then Q," and "if Q then P."
Another way to say this is that P and Q are EQUIVALENT.
And finally, to spell it out in all possible ways,
what we are really saying
is that P and Q must BOTH be true, or P and Q must BOTH be false.
All these lines mean exactly the same thing.
In casual English, when we are not doing math,
we are, well, quite casual with the word "if".
Sometimes when we say "if," we mean "if";
but other times, we actually mean "if and only if."
We CAN'T have that in math.
In math, words must have very precise, unambiguous meanings,
and that's why we now have the two terms:
"if" on the one hand, "if and only if" on the other,
and we have symbols for both of them.
To conclude this video, I have one more little test,
and this one is going to be a bit counterintuitive.
Here is an exercise: is this true or false?
If 0 is greater than 1, then this number is prime.
Pause the video for 10 seconds, decide what you think,
and then continue watching.
Let's look at the two parts separately. Well this is the easier one, right?
Zero greater than one? Mhmm, we know that's false.
But what about the one on the right? Is this number a prime?
To be honest, I don't know.
It's odd, so that's a good start, but I can't quite tell at first glance;
I don't know if it's a prime or not.
So what about the whole implication? Is it true or false?
It turns out, I don't need to know whether that number is a prime
to conclude that the FULL statement must be true.
And why is that?
Well, remember that "the implication being true" means:
if the "IF" part is true, then the "THEN" part must be true as well;
But if the "IF" part is false, the "THEN" part can be anything - we don't care!
Whether this number is a prime or not, the "IF" part is FALSE.
And, therefore, the WHOLE implication is TRUE.
I can conclude that without knowing whether this number is a prime or not.
Perhaps this is a bit counterintuitive,
but if you fully understand these examples, and you are comfortable with them,
then congratulations!
You now understand how we use conditional statements in mathematics.