Which two of these is not a topology on X = {1,2,3,4,5} .Select one or more:a. {0, {1,2,3,4}, {1,2}{1,2,3} }b. (X, {1,2}, {4,5}, {1,2,4,5},0} cX,0, {1}, {1,2,3,4}} ...

Which of the following pairs of sets are disjoint (i) $\{1,2,3,4\}$ and $\{x: x$ is a natural number and $4 \leq x \leq 6\}$ (ii) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$ (iii) $\{x: x$ is an even integer $\}$ and $\{x: x$ is an odd integer $\}$

In this problem. What we want to determine is which of the following is not an arithmetic sequence and we're going explain why? So it might be easier to figure out which ones were Well, if it's an arithmetic sequence, that means we're adding a constant to get from one term to the next. So let's start with part a part. Ay, we have 1/2 1 3/2 Well, to get from 1/2 toe one, we add 1/2 and to get from 1 to 3/2, we add 1/2 so d is equal to 1/2. Therefore, this is an arithmetic sequence I'm going to skip be for right now. I'm gonna jump to see well, to get from 5 to 0, we add negative five and they get from zero to negative five. We also ending in a five so d would be negative five. And therefore, this is also in arithmetic sequence. Well, let's look att de to get from to the three, we add one and in front three before we had one so d would equal the one. So this is also in arithmetic sequence. So your guess is probably right B is not. And the reason why is well, how do we get from 1/2 to 1/3? They make it sound like you just simply add one into the denominator. However, that's not true. So if I needed to find D, I would need to do 1/3 minus 1/2. Well, were subtracting fractions, which means we need a common denominator, which in this case, is six. So I'm gonna multiply my first fraction by two over too, and the second fraction by 3/3. So this is going to give me to over six minus 3/6. Well, to over six minus 3/6 is equal to negative 1/6. So that would be our common different. Some subtracting 16 Well, that would mean that if I do 1/4 minus 1/3 I should also get negative 1/6. Well, let's see, while our common denominator at this time would be 12 so I'm gonna multiply my first fraction by 3/3 in the second fraction by four before Well, that means our first fraction will become 3/12 and their second fraction will become for over 12. Well, 3/12 minus former 12 is negative. 1/12 and negative 1/12 is not equal to negative one over sex, which is why part B is not an arithmetic sequence.

Because of the finest function is well, you can X comes along with the axe X minus one Cancer with Agnes one about to hear. Counselor, we with this one and express one with the experts One will be canceled on us. Well, this about tree counselor would hunting here. So we have left here will be ex minister on the top. On the bottom we have the ex ministry so the function is continues except for extra coaching tree. So the question here asking which under foreign point Ah discontinued contingent is not removable So the answer I have a bit of a ex equity tree will be not removable.

Which of these collections of said our partition beset us? Just one Q B 456 So the definition of tradition Yeah. Oh, sometimes things like this. It's not the union of these two subsidiaries. Not 12 before, but that 1 to 2. The fourth That's gonna have repetition. I'm like, yeah, sorry. Said cannot have repetition. I'm like, so you if you take the union of old, you get one Q two, 34456 So during up in this joint. Therefore, they don't partition here to set here. And you're not this joint because they share too in these to share for now for the other set in part B. We're getting set the collection one. And to make this simpler rain dick early recent for so I'm just gonna right, I'm just gonna list out elements in the set here were given one who were given 26 and were given 45. Well, if we take the union and quality said we doing in fact, if we take the unit of 1236 four following we yet 1236 the union before in the Unit five, which is just for five we get 12345 You can rearrange position because it said doesn't matter which indices not and then even us. That means this collection of partition now support eat were given 246 and 135 Well, already, you can see that there is no common element inside. I hear that. And because they contain all the elements inside, they did do petition. The bigger said before importante were given 145 and to six. Well, here nobody sets contained elementary. So when you take the union of these sets usually get 1 to 4 56 He's not inside the set, so that implies you're not a petition to set.

So for this question and seeing which of the following is true, But the graph of F X equals excellent for about five exits. Zero little choice. So it's either a corner, a cost of vertical tension or dis continuity, or it doesn't exist if there's no exist. So we've already dealt with problems that half of X equals, except for fifth. So if we remember from those with the answer is, then we're good to go. But to explain it one way that we can do this. There's Coolidge Window, where we can first graph it if we graph it. We see we have something that looks like this. We can go ahead and say, Yup, that's a cusp. But what else we can do is we can take the limit of ex approaching zero from the right side versus a limit of ex approaching zero from the left side of how we would take a derivative. So we have X minus half of a, which is looking zero in at zero over X minus a so ex same thing here B f of X over X, and this would give us one over X to the one over five. If we just get it out the right side, this is going to give us a positive 30. If you go from the left side, this is going to give us negative infinity. So we get different things when you're approaching from different sides. So that means that when we have a of expose except for fifth is not defensible at X equals zero. Because there we have a custom that I do it.