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Proof that: If B ⊆ A, then P(A − B) = P(A) − P(B); in particular P(B) ≤P(A)....

Question

Proof that: If B ⊆ A, then P(A − B) = P(A) − P(B); in particular P(B) ≤P(A).

proof that: If B ⊆ A, then P(A − B) = P(A) − P(B); in particular P(B) ≤ P(A).



Answers

Suppose that $A \subset B$ and that $P(A)>0$ and $P(B)>0 .$ Show that $P(B | A)=1$ and $P(A | B)=P(A) / P(B)$.

So if a and B have the same card in ality, that means by definition that there's our baj active map that goes from a to B. Orin, it's 1 to 1. Now let's define ah function G that goes for on the power set of a to the power set of b their maps A subset s of a to the side fos which article is defined as the set off f of little A such a little a easiness. So is the image off all the elements off the set s We want to show that these g the fines objection and that will show that the community off power set of a is equal to the originality power set of me so easy injected. So let let's say that G off big axes called tau G off big y. That means that by definition, off g the set containing all the images off the element of Big X is equal to the set off the images off all the elements off big y. But now this these two sides are equal. That means that for every X, there is a why such that f of X is equal to half. Why? Because the first site contains all the images off little acts well of all the possible. It relaxes, and the second set contains demons of all the possible little wise. So there must be a projection between these two sets, in particular our effort. Back physical tough of wife for some choice of X and y. But now F was objective. So X must be equal to why and these works for every element X and every M. And why in a big exam big. Why so upset? Big X is equal to the set Big Y and therefore G is injected. Then it's just subjective. So let's take big set Zed in power set off me. We want to find something, empowers it away that gets mopped to that. Now that is a subset of being, by definition, off our side of me. Let's put X to be the counter image through F, Off said, and I called. It is, by definition the set of the country images off, Little said. We're sad ranges over set, Big said. He said ex lives inside A and by construction. G. O VE. Jax is exactly that which shows that G subjective. They're for G is both injected and subjecting, which means it's a projection

Hello there. In this case we need to prove the following statement. Let's say that the power set of A is contained power set of B. If and only if the set A. Is contained in. So here we have a double implications. So we need to prove for each direction. So let's start with these directions here. So if we want to prove in this direction means that we need to assume that the power set of A is contained in the power set of B. And in some way we need to reach this that the A. He is contained in the set. Okay, So let's look at this in the following way for all the elements in A. Okay, we can form a single song that contains only that element. Right? This upset that contains only the element A. And this is a subset of A. Okay. And if all these subsets, all this possible subsets are all these elements here are subsets of A. That means that these elements here, he said, is an element of the power set of A. Because the power set of A. By definition contain all the possible subsets of A. And or hypothesis, say that the power set of A is contained in the power set of B. So that implies by the hypothesis here that this element here is also contained in the power set of B. And if this singleton is contained in the power set of B, that means that the this set is a subset of B. And even more this implies that the element A is contained in B. So from this we need to focus on this part. So technically from all these implications, we have that for all the elements in A. That element is also part of the city. And this is equivalent to say that this is A. Is contained in the settlement or is a subset for the other direction. We need to consider now that A is a subset of being and we need in somehow we need to reach that that this implies that the power set of A is contained in the power set B. Okay. So we need to prove this. Okay, so let's consider that. See is some subset of a an arbitrary subset of a by trans activity of the inclusion here of the subject. Operation here, we have that. And by the hypothesis, of course this we have that C is also contained in the sector V. And this is any subset of a an arbitrary subset of pay. So these, what I have described here is equivalent to say that the elements C is uh this upset C is an element of the power set of A. Because by definition the power set of A contains all the possible subsets of, and we have that this element C is also a subset of B. That means that this subset C is also an element of the power set of the and this is equivalent to say that the power set of A is contained in the power set of, and with that, with the.

In this problem Where us to prove April the easing of a private seat When, uh, Sethi and see our ego. So let's say that element A is insects, see, And some reading orbiter Element X is in such B So from this, we know that a s will be in port. Eh? Be now I assume a protectee is equal to a C. Then this means that eggs we'll be in a what I see Hence element X we'll be in said See Right. So if he is, you hate to be a fix. Uh, element here eggs isn't returning elements in your victory element X That is instead be we found that it will also being set seed. So this means that B is a subset of C No, this time, let's use April showers. Let's say that again, eh? A. Some Biggs elements of settlement is an element of Hey, now it's some run random arbitrary Ellman is an Elden offset. See? So from this we know that airs will be a said all by a former C. Again, let's assume a product be to be equal to a vaccine that from this we see that a X we'll be in. Hey, brother. See? Right. Some pens. Thanks. Sorry. We already started with a C. So that should be it for me and x will be inset Be so again X could be any element. So this means that any element that's he has he should also has that. So this means that sea is a subset of the if bees the substance e and it sees a subsequent feet This means that being and see our people So this means that product baby is evil for they see it says B and C are people.

In this problem We wanna short of aces. If development is a subtitle A compliment. So we're gonna issue We're gonna start by assuming this to be true so soon Hey, is the substance of the then for every ex that is an 11 to pay x will be in a lento Be pricing, Cesar. Subset of feet. Now we know that this is true. Yeah, for everyone, that is a l ittle b Pullman. This implies that it wise and element will be a compliment. Then why is not an elemental said right? And since be known at, um, for every X that is an elemental fight, it's also the element be You can say that. When will I be that it's not an elemental Hey, right. So this is basically equal saying that it was an element of Republicans. Then why's an elemental eh? And since this is true for each and every wire, since that is true for oh wise, this means that be compliment is a subset of foam. So we just showing that if be compliment is a subset of a compliment than eight is a subset


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