Right, so in this second video, the first thing I'm going to do is to recap the symbols that were in the last video, because it's a good idea to see them again. Then I'm going to go on to show you a new one. Then I will say something about implication and the nature of it. And finally, we will...
do some calculations with truth tables, which are very useful as exam questions. And that's a hint, by the way. So, last lecture, we had propositions.
P, Q, maybe R could go on. We're using these letters for them. And the first way to connect them together was this, which is and.
So P and Q are both true. And we saw the disjunction. That's P or Q is true.
One of them can be true. Both of them can be true. But the only thing you can't have is that both of them are false. Then that is false.
We had... And... The exclusive OR.
So this is where P is true or Q is true but not both of them. We saw negation. So this is not P. So if P is true, not P is false.
P is false, then P is true. Then we had the conditional which can cause some... problems, but it works like this. If P is true, so you only have to look at it when P is true, but if it is, then it needs to be the case that Q is as well.
So this is always true, except in the one situation where P is true and Q is false. Okay, so we're going to add to that. the biconditional and write it P with a double headed arrow and This is to signify that these are true exactly when the other ones are so we write this as P if and only if P if and only if Q. Okay, so what should the truth table of this be?
Well, we're defining it, but we know what we're aiming for. We want it to be true whenever these match. So as long as these match, we're happy.
But if they are different, then we know that they are not the same. They do not have the same valency. Okay, so that's the biconditional. And it turns up in maths in a number of different ways.
Sometimes two statements genuinely are equivalent. So if x is 3... That is true if and only if x plus 1 is 4. And that means if I start from here to go here, then this will be a true statement.
So if I know that, then this will follow. But it goes backwards in this case. So if I know x plus 1 is 4, then I can take one off both sides and I can get to x is 3. So it goes both ways.
Let me give you an example. where it doesn't. So Q implies P is called the converse of P implies Q. And it doesn't always hold.
And it doesn't always. Well, let's say it doesn't always. follow from p implies q so you don't always get a double-headed arrow. You may only get it one way.
Let's take the example from the notes, just expand on it a bit. So, e.g., we're going to take p equals x. x is bigger than 2, and q is the statement x squared is bigger than 4. And it is true that if p is...
If p is true, if x is bigger than 2, then x squared is bigger than 4. And let's just see why. So we're going to assume x is bigger than 2. That's our starting point. And then, so let's assume this. Since x is bigger than 0, because they're equal to zero even, which is a trivial consequence of this, then I can multiply both sides of the inequality by this number x, and it will stay true. So if you have one number that's bigger than another one, and then you just scale the whole thing, or shrink it, it will still stay in its relative position.
So we have... x bigger than 2 implies multiplying both sides by x. We have that. And another way to write that is x squared is bigger than 2x. But also, x is bigger than 2. So if I take x...
and I swap it for a smaller number, then this multiplication will go smaller. Another way to think about that is if you multiply this equation by 2, then you get 2x is bigger than 4. So this is 2 times 2, and this is 4. So this is Q. So P implies Q. Very good.
What about the other way around? Let's assume Q is true. X squared is bigger than 4. Must it be the case that X is bigger than 2?
The answer is no, because negative numbers, if you multiply them together, become positive. So Q implies P is false because, well, I only have to give you one counterexample. So X is minus 3 implies X squared is 9, and that is bigger than 4. But this is definitely less than.
two right okay i think we will leave the truth tables to the third video