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Suppose that R is the following relation on the set A = {1,2,3}: R = {(1,1),(2,2),(2,3),(3,1), (3,2)} Determine which of the following is the relation R2R1. Rl = {(...

Question

Suppose that R is the following relation on the set A = {1,2,3}: R = {(1,1),(2,2),(2,3),(3,1), (3,2)} Determine which of the following is the relation R2R1. Rl = {(1,1),(2,2),(3,1),(3,2),(3,3)} R2. R2 = {(1,1), (2,1),(2,2),(3,2)} R3. R3 = {(1,1), (2,1),(2,2),(2,3),(3,1), (3,2),(3,3)} R4: R4 = {(1,1),(2,1),(2,2),(3,1),(3,2)}

Suppose that R is the following relation on the set A = {1,2,3}: R = {(1,1),(2,2),(2,3),(3,1), (3,2)} Determine which of the following is the relation R2 R1. Rl = {(1,1),(2,2),(3,1),(3,2),(3,3)} R2. R2 = {(1,1), (2,1),(2,2),(3,2)} R3. R3 = {(1,1), (2,1),(2,2),(2,3),(3,1), (3,2),(3,3)} R4: R4 = {(1,1),(2,1),(2,2),(3,1),(3,2)}



Answers

Show that the relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5\}$ given by $\mathrm{R}=\{(a, b):|a-b|$ is even $\}$, is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other. But no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

In this problem of religion and function we have to show that relations are is equivalence religion and relations are is given in the set A. Is a set In which X belongs to shed and such that X is a value between zero and 12. So this is the set A. And relations are is given way. Say relations are. Is this is the first. So relations are is given by R is a relation between order to pay A and B. Such that models of a minus way is multiple of four. Is multiple lawful. So this is the relation. Now we have to show that this is a equivalence relation. So here's A. is the value between 0-12. So we can digest is a set in which the values are from 0 to 12, like 0123 Up to this is 12. And now we can write our so our would be see this should be a multiple of four. So first we are taking four and zero the first wearable before and zero. So we can write it four and 0. And also we can take zero and food. So this would be zero and 4 and now 15 So this would be 15 and 51 and then 266, 2, 26 and 62 Similarly this would be Up to 12 9, 9, 12. Yeah 12 and nine and 9 12. And then this will be Here 844, 8, 84 for it. And in the end this would be 12 -12 or we can say 12, 12. So this is our and now this relation is said to be reflective if ordered pairs like X and X should also be there so which belongs to the given are so when we write X is equal to 0 to say we are saying 4567-819. So there will be always such bear. Such as this is like 7 -7 would be equals to zero. So this is a multiple of food. 8 -8 models of 8 -80. So we can say that relations are is reflexive. And now we have to check for Senate regulation. So suppose the ordered pair A. B. Is in our then ordered pair be and should also be there. So as you can know that their 40 is there that consider four is also there one and five is there five and one is there to and six are there and six and two are there. So we can say that our is symmetric. And now we have to check for transitive. So transitive relations says that transitive relations says that if A. And B. So here if A and B. Are present in our and another pair BNC is present in our. Which implies that there should be any pair which we have A and C. Element should also be there. So now let's take an example. So suppose we are taking an example zero and four, so we here we are taking zero and 4 and models of zero and food is equals to this is the models of zero and 4 Is equal to four. And now BNC say this is four and 8. So say this is four and 8. So models off, this is 4 -8 is equal to food. This is also a multiple of food. This is also a multiple for both are in this set. This implies that the ordered pairs zero and see that means zero and 8 should be there. And now we have to He has satisfied the condition that means models of 0 -8 is eight and 8 is a multiple off for so we can say that our S transitive relation since this is reflexive, symmetric and transitive all together. So we can say that our is hence odd. Is and equivalence relation are is an equivalence relation. So this is the answer. And now we have the second part. So 2nd part says that here are is a relation. So our as a relation between ordered pair A and B. Such that is equal to be so it is equals to be. So now when we take the value is equal to be so that's what we it is equal to be. So that means we can write A. Is equal to A. And that means the ordered pair X and X A N. A would be a part of our. So we can say that our is reflexive. So art is reflexive and now when X and a here X and X. R equals and A and a R equals. So that means the ordered pair would be of the form of X and X. And now when this is converted, so this will be again ordered pair X and X. So this is the condition for a relation to be symmetric. So this will be a symmetric relation also. And now so we can say that our AIDS symmetric also. And now we have to check for transitive relation. So a relation is said to be a transitive if say here A and B. R equals. So that means we have the value X and X. So X and X. And when we take the value between 0-12, X would weigh from 01 to up to 12. So the pair would be of the form of severe taking 00 So this would be zero equals to zero. That means this will be 00 and another value should be equals to 00 So another value will be 00 And when we write the ordered pairs zero, this zero and the zero should we also present there? So this is this would be a part of this set are also so we can say that this is a condition of transitive relations. So we can say that our is transitive since this relation is reflective symmetry insensitive all together. So we can say that hands our is and equivalence relation. So are is an equivalent relation.

So somebody problem? We will be confirming assuming the composition of do re nation are Rick itself. But first we will represented regulation using with you were warned me trip. No, we have What we will get here is a five by five metric king for the first rule, there's only 13 so it will be narrow, You know 100 Uh, second line, second roll. There's only two horse heroes. Here's one there. King. Third liners. 3135 We have wanted him on. And poor girl only the third position the one And with fit through the birth Second on the point that they can have one. So this is this the wrong matrix here, Nor in or a we're finding, uh, weird. So with this, this would be the composition of our it felt. Which means we're going to do the bully and product of only drink and so are with okay When we multiplied without, we would do the bullion multiplication We end up with a matrix But this hair in run So and, uh get the matrix one Well and born, there was no one who was ill. I'm assuming here you already know how to do the bullion product. You can review section 2.6 in your book if you're not, if you do not remember Oh, for more You're okay Based on later something r squared. I know First rule We got the point worn one in 15 beckon room We have 2.3 under third world Get for you one Great to hurry Hurry! And 34 under four through we have for one a report by on under panel Will we have by four Very five five Knowing part B we are finding are so cute And that would be the composition of our are or the bully in front of over Answer in part One part A Sorry. So let's just call it quits. Here, boy, use the bar. Okay. The ones we actually do the bullion product What we get for our you under first year of the Matrix do you have one? Why? To 13 Anyone for under second you have to 1 to 5 under third rule we find where you wan 33 And for I'm 35 under four through or one or two for three and or or I'm the final girl. You get points five born 53 and 55 now for the remaining cards were essentially pulling the team that still in part C you're finding are to the fourth and again does the composition of our with our answer from the previous part The Matrix from part C. And we do the bullion product with and so are so one that computation is done. The resulting matrix leaves us with the following points From the first room we have 11 13 14 and 15 Respecting room. We have to one to to three into four. So these are recalled is all the positions where there is a one under me in the Matrix on the Third World we have no one. Me too. Very very 34 and 35 14 Throw you have Or one for three, or or and for fi from the final room you have are born I to on three by four and by five. So these are outpoints from the relation off. Four Our to fourth Well, part D started a bit by now you get a pattern. So we're going to do and so far to the fifth bullion producto with and are from the resulting matric. What we get from the first rule. Yeah, the whole entire row is one. So we have all the points 11 all the way through to 15 In his second rule, you have to one So three before into five in the third row they're all once so I'm 31 all the witches to five. They replies for before my warrant for one, all the waiting for fire I know a fit through is also all one. Well, here we go from by one only way to by five Is the party knowing party we're finding are to the sick Based on the pattern, we know that this is the major and some are to the fans. I will take the bullion port up with EMS of our once we can do computations. What we end up with is the five by five matrix That is where all the entries are equal to one so essentially are to the six would be would give us 25 points. So for the sake of saving time, the first world will go from 11 all the way to 15 That general goes from 21 on the brink 25 There you go. And I was asked for Was this name no, by born all the way through five. But doing all we should have 25 points if you read it all out. But if this war card and in part we're being asked to generalize hard to the end on and based on our answering party, we can see that are to the end would be the same Arjun as a loner and is grated under equal. Okay, so for F or on onwards, the answer will be exactly the same as part.

Were given to relations. We were asked to find the composition of these relations. Relations are, are which is this set? 1213 to 3 to 431 and s which is a set of work in pairs to 131 32 for two. So we have that s composed with our is going to be First we look at first elements in our which is the ordered pair 12 and then we see in s which ordered pairs start with two. So that s has to one. So one of the repairs is going to be one one and that's the only ordered parent s. It serves to moving onto the next order parent or this is 13 So now we're looking for order pairs and s to start with three. We have 31 is another ordered pair so we get 11 again. Must have 32 is nor compare anus. So we get one too. In s compose are That's the only other order Parent s starts with three moving on to the next Working parents are we have 23 and again there to her ordered pairs and s to start with three. There's 31 so that we get to one ness compose are and three to so that we get to two in s composed with our and never moving on to the next word pair and are yet to four. And it s there's only one or repair starting with four, which is four to So again we get to two in s composed with our and finally the last week, apparent artist 31 And we see that s there are no ordered pair to start with one and so s composed. Are is going to be 1112 to 1 to two.

In this problem we have given that let R be the relation in the set. So say this is a set which have elements 123 and four. Is given way the relation R. Is given way message 12 2211, this is two 11 44. This is 44 1333313, then 33 And then 3 2 trickle. And now we have to find the type. So first we will check for reflexive if the ordered pair say X and X. From here belongs to R. And X belongs to the given set A. So it could be 1234 So ordered pair 11 two, 33 and 44 should be in art. So these all belong from our because 11 two, 33 and 44 present. So we can say our age reflects you. Our is reflexive. And now we have to check for symmetric. So here 12 belongs from our And then which implies that too. And no one should also in our. So so check for two and one but there is no two and one. So we can say that our is not symmetric and now we have to check for don't do So if the audit pair went two or 13 say So we are saying 1 3 belongs from our and another pair 32 which is here belongs to uh huh. Which implies that the order paper one and two should be in R. One and two should be in R. If the season are this will be submitted, this would be transitive. So now we have to check for it 12 is present. So we can say that our is transitive relation. So check for the option. So if our is reflexive and transitive but not symmetric. So hence option B is correct. So we can see that our is odd is reflexive and transitive and transitive, but not cemetery. Hence option B is correct. This option B is correct, so this is the right answer.


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