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3 A relation R is defined on Zby aRb if |a-bp 2: Which of the properties reflexive, symmetric, antisymmetric, and transitive does the relation R process? Justify yo...

Question

3 A relation R is defined on Zby aRb if |a-bp 2: Which of the properties reflexive, symmetric, antisymmetric, and transitive does the relation R process? Justify your answer.

3 A relation R is defined on Zby aRb if |a-bp 2: Which of the properties reflexive, symmetric, antisymmetric, and transitive does the relation R process? Justify your answer.



Answers

If $\mathrm{A}=\{4,6,10,12\}$ and $\mathrm{R}$ is a relation defined on $\mathrm{A}$ as "two elements are related if they have exactly one common factor other than $1 "$ ". Then the relation $\mathrm{R}$ is (1) anti symmetric. (2) only transitive. (3) only symmetric. (4) equivalence.

In this problem of relation and concern, we have to check whether the relation is reflexive, symmetric or transitive and relation is art. Is in are they twins relation, agent? The set of real numbers and relations are is given as relations are is a relation between order pair A. And B. Such that is less equals than BQ. And now we have to check. So first they will check for the reflexive. So a relation is said to be reflective when the ordered pair A and A which belongs to our. And now if A N. A belongs to this relation are this must satisfy the condition that is is less or equal to its cube. So now let's take A. Is equal to four. So which means that food is less equals than XQ, which is true. So this would be 64. And now when we take the value between zero and one, so say 0.1. So this says that 0.1 is Less or equal than 0.1 Q. which is 0.001. So which is false. So we can say that this relation R is not reflexive. And now we have to check it for symmetric. So now symmetric her relations says that the ordered pair say A. B. If a B belongs from our it says that B. And A. It say that also belongs from art. So now let's take support and we belong from art. So say is equals two, Say a is equal to 10 and B is equal to three. So condition will be 10 is less than three Q. Which is 27. This is true. and now this condition should be satisfied that it be is here three is less equals than he has a cube. That means this would be thank you here which is again through but But when we take is equal to one And say B is equal to three. So this will be here. One is less than three cube, which is 27, which is here too. And now When we write it here so ordered pair three and one should also belong from art. So this would be now the condition would be three is less than one cube which is one. So this is a false statement. So we can say that our is not symmetric and now we have to check it for transitive relation. So so now we have to check it for transitive relation. So a relation is said to be a transitive if the ordered pair A. B belongs to R. And then also B and C. Which belongs to our, which implies that the ordered pair A N. C should also belongs to our. So now let's take A is equal to see if we are taking seven and B is equal to two. So condition will be is less than two cubes. So this will be seven is less than it. It is true. And now we are letting C is equal to say 1.5. And now the other pair would be Here B&C. So this would be two and 1.5. So the conditions would be Two is less than 1.5 Q. Which is equals to 3.375. So this is again through. And now this should must satisfy that means A. Is equal to seven and be here see is equals two Say 1.5. So when we write it, the condition is should be less than BQ. So this is seven should be less than 1.5 cubic feet equals to 3.375, which age falls. So this is false. So we can say that our is not transitive relation. So we have checked for reflexive, symmetric and transitive

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.

Hello students. And this question we are given to persons. And this question we have given to persons A. And B. And we are given that A. Is related to be if A. And B have same brother. Okay. And we as let it if they have same brother. Okay. Now we need to tell whether this relation is a reflexive, symmetric or transitive or equivalent. Okay, so faster fall. We'll check for reflection. Okay. So checking for reflexive relation. Uh if a function is reflexive, then it says that A should be related to A. Okay, So if we see is ready to A. That of course is ready to A. Because a N. A will have same brother. Okay. Same person can will have same brother. Okay, so this function is reflexive right now we're checking symmetry, checking for symmetric then not that in symmetric function. It says that if A is related to be then we should be related to a. Okay, so if you see a and we have same brother, then it is obvious that b. And they will have same brother. eight. So a. No, no need to write the full statement. It is also cemetery function symmetric regulation. Okay. Now checking their transitive transited and transitive, it says that if A. Is related to be and B. Is related to see then A should be related to. See. Okay, this is the definition of transitive. Now, not that. What is the uh what is the relation are if A. And B have same brother. Okay. And B. N. C have same brother. So it obviously implied that A. N. C will have same brother. Okay. Because they are the siblings. Okay, so this function is also transitive. So we can say that this function is an equivalent relation. Okay, This function is an equivalent relation. Because what is equivalent relation? The relations which are transitive, symmetric and as well as a reflection. Okay, So fourth option is the correct option. Thank you.

In this problem of religion infants. And we have to show that the religion are in the set set is which has the elements 1, 2 and three. So and the relation is given us given by relation is given by this is only two other pair, one and two, and the second one is two and one. So this is the Under repair which have only two pairs that it went 1-1. And we have to show that this relation is symmetric but neither reflexive nor transitive. So first we would So for reflective. So relation is said to be reflexive if ordered pair A and a part of our so ordered pair. Eh and S say when we are putting one, So this should be 11 When we are putting too. They should be too. And when we are putting three, the should be 33 And this will be true for every values of A. Which belongs to the given set A. So here we have three elements 123 but there is no such pair. So we can say our is not reflexive. And now we have to check for here symmetric relation. A relation is said to be symmetric if ordered pair even and a two which is already in relation are which implies that 8 to even should be prison there for every values of even and 82 which belongs to the given, said A. So here we have the give and take 812 and this is a pair. 12 so 12 is in relational. This implies that to anyone should also be present there and this is also present there. So we can see that this relation is a symmetric relation. So R. S symmetric relation are asymmetric. And now we have to check for transitive. So again, if ordered pair X. Y. Which belongs from are they want to? And another order pair. Why end Said which belongs to here. I say this is two and 1. This implies that the ordered pair X. NZ should be also presented there. So that means there should be one and one pair, since this is not present there. So we can say that our is not transitive, so are is not transitive. So hence we can say that we have proved that R. L symmetric, but neither reflexive, not transitive. Hence odd is symmetric, but neither reflexive. Yes, neither reflexive, not the north transitive. So this is the proof.


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