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Exercise 1. Consider the binary relation on N X N (which is a subset of (N x N) x (N x N)) defined by (a,6) (c,d) if and only if a+d = c+b. Prove that this is an eq...

Question

Exercise 1. Consider the binary relation on N X N (which is a subset of (N x N) x (N x N)) defined by (a,6) (c,d) if and only if a+d = c+b. Prove that this is an equivalence relation. You may only use properties of N, not Z_

Exercise 1. Consider the binary relation on N X N (which is a subset of (N x N) x (N x N)) defined by (a,6) (c,d) if and only if a+d = c+b. Prove that this is an equivalence relation. You may only use properties of N, not Z_



Answers

a) Let $R$ be the relation on the set of functions from $\mathbf{Z}^{+}$ to $\mathbf{Z}^{+}$ such that $(f, g)$ belongs to $R$ if and only if $f$ is $\Theta(g)$ (see Section $3.2 ) .$ Show that $R$ is an equivalence relation.
b) Describe the equivalence class containing $f(n)=n^{2}$
for the equivalence relation of part (a).

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.


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