TMUA: do you truly understand "X implies Y"?

[00:00:00]Today I'm going to be solving a very nice TMUA style problem created by J Z Maths. If you want to see the full paper, go on to my Instagram, make sure you're following me, and comment the word nine, n i n e, under my latest reel, and I'll send it over to you.

[00:00:13]Let's have a look at this problem.

[00:00:14]Really, really nice. This is from paper two of a TMUA style paper. Let P be a property of the real number X. Each of the following is a statement about real values of X. Exactly one of the following statements is true. Which one is it? This is a classic type of problem where at least TMUA logic style problem, where we've got to kind of use logic to help us out here. The idea to solving this is you essentially want to go through the statements and ask ourself if that statement was the correct answer, if option, let's say A was the correct answer, would that then imply any of the other statements are true? If the answer is yes, we can eliminate A because we know that exactly one of these statements is true. So, if A being true also meant, I don't know, C was true, then that means not exactly one of the statements are true. So, therefore A can't be the true one. And we can kind of go through one by one until we get the answer. Obviously, when you get good at these sorts of things, you can maybe think about other things here. But in particular, one thing I'm noticing is that these two statements look kind of similar. These two look kind of similar. These two look kind of similar. And then you've got G as well.

[00:01:15]Anyway, let's get stuck into this one.

[00:01:19]So, we're going to start with statement A and say, "Well, if modulus of X is less than one, then P is true." And I'm in particular looking at how it relates to statement B here as well. Statement B says, "If the modulus of X is less than two, then P is true."

[00:01:33]And perhaps maybe we can notice that, well, B can't be the correct answer.

[00:01:38]Because if, again, if B was the correct answer, so if the modulus of X being less than two implies P is true, so imagine that was correct, then the modulus of X being less than one, well, if the modulus of X is less than one, then for sure the modulus of X, oopsie daisy, is also less than two, but we're assuming that that means P is true.

[00:01:59]So, therefore, the modulus of X being less than one also means P is true, and therefore option A would also be true if B was true. And therefore, we know that B cannot be the correct answer.

[00:02:10]So, if I get rid of all of this, we can say B is not correct.

[00:02:14]Easy peasy.

[00:02:16]Let's get rid of those just so I have enough space here.

[00:02:19]So, B is not the correct answer here.

[00:02:20]Okay.

[00:02:21]Let's have a look at C and D as well just because those look interesting. Um if P is true, then modulus X less than one, and D is very similar but modulus X is less than two. Using this again, we can eliminate C because if C was true, then that means P being true implies modulus of X is less than one.

[00:02:40]But, if we look at D, P being true according to what we know or what we're assuming is true because we're assuming C is the correct answer, that would mean that modulus of X is less than one, but one is less than two.

[00:02:52]So, the modulus of X is also less than two. And therefore, D would also have to be true. And since exactly one of the statements is true, we can't have C as true.

[00:03:01]Okay, nice. So, we can kind of go through this like so. Let's look at the next kind of pair of similar looking elements. So, E and F, if P is not true, then X squared is less than or equal to or bigger than or equal to one. And F, if P is not true, then X squared is less or bigger than or equal to four.

[00:03:15]Now, this is a little bit topsy-turvy.

[00:03:17]So, what I'm going to do here is use a nice little trick. I'm going to write down the contrapositives of these statements. You don't have to do this, but this is just more of an excuse for me to explain what the contrapositive of a statement is. So, if you have a statement X implies Y, the contrapositive of this statement is not Y implies not X. And these two statements are equivalent. So, here, this statement is equivalent to if X squared is less than one, in other words, modulus of X is less than one, then P is true.

[00:03:47]And then this one is if modulus of X less than two, then P true.

[00:03:54]Okay, cool. And this looks very, very similar to both statements A and B. And so, in fact, since statement E is the exact same as A, two birds with one stone, we can eliminate those. And statement F is the same as B, so we can also eliminate F as well. Very nice. So, now we're left with D or G.

[00:04:10]Okay, well, let's look at G G. We haven't looked at that one yet. P is true only if X squared is less than one.

[00:04:16]Remember, X only if Y means X implies Y. So, here we're saying P implies X squared is less than one.

[00:04:27]Um or in other words, modulus of X is less than one. Um and that's the same as C. And so, therefore, we can eliminate G. And by process of elimination, D must be the correct answer.

[00:04:39]Really, really nice problem. Let me know if you solve this in the comments down below. And stay tuned for more videos like this where I solve TMUA style problems on this channel. I know a lot of you watching this are preparing for the TMUA, so I'm going to be making lots and lots of videos which are going to be useful for you in your preparations. And if you're looking for some extra support, I do help students as well.

[00:04:57]Over 83% of my students get offers from either Oxford or Cambridge. So, if you're interested in getting my support, I'll leave a link in the description below, where you can book in a call with me, and we can see if I can help you get to your goal.