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Let A, B and C be sets. Show thatTnBnc=Tu CUB b) Au(Bnc) = (CuB)nt...

Question

Let A, B and C be sets. Show thatTnBnc=Tu CUB b) Au(Bnc) = (CuB)nt

Let A, B and C be sets. Show that TnBnc=Tu CUB b) Au(Bnc) = (CuB)nt



Answers

Let $A, B,$ and $C$ be sets. Show that $(A-B)-C=$ $(A-C)-(B-C)$

Hello there. So in this exercise we need to show that if is containing B and B is containing C, then A is contained in C. As well. So this is uh technically this the the the trans activity property of the of the subsets. Okay, so first let's illustrate this with the band diagram drawings always helps to show the things. So let's drove first this part. So there is a set of V. That is going to come, there's going to be the set V. And there is a subset of be, that is the set A. But this part of the statement said that B is a subset of a set. See that means that there exists another set that contains B. And you can see just by this illustration that A. Is also going to be contained in C. In the words in writing here, we know that the meaning that A. Is contained in B. He is equivalent to say that for all the elements A. In this set A then A is also an element of B. Okay, so this is for all the elements in A. But also we have this statement here that say that B is a subset of C. There is a cool way to say that for all the elements B, B, then B is also an element in the set seek. But here we have that all the elements of A are elements of B. And here we say that all the elements will be our elements of C. So therefore we have that all the elements of A are going to be elements. Oh, see mm.

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

Okay, so in this question, we want to prove that a is a subset of B if and only if being confident is sub set off a comet. So we prove this one first. So let's a b a subset of B then if X is, it is not be these implies that X is lost in bay. This is pretty obvious because if you have set B and you have a set a, well, just contained inside of it, even element is outside of big, then it must be outside. So using definitions off compliment if exits and be this implies that X is in a compliment. So therefore, being confident is a subset of a compliment. Now in the other direction. Well, let me compliment is a subset off a in. Similarly, if an element is not in a compliment, then it follows. Our eggs must not be in his confidence. So therefore, if X is in a a, then X must be in B Therefore a subset puppy

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.


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