(a) Let f be a function from A to B. Define the relation T on A by x T yProve that T is

Chapter 4, Problem 18

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QUESTION:

(a) Let f be a function from A to B. Define the relation T on A by x T yProve that T is an equivalence relation on A.(b) In the case when is given by describe the equivalenceclass of 0; of 2; of 4.(c) In the case when is the cosine function, describe the equival

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QUESTION:

(a) Let f be a function from A to B. Define the relation T on A by x T yProve that T is an equivalence relation on A.(b) In the case when is given by describe the equivalenceclass of 0; of 2; of 4.(c) In the case when is the cosine function, describe the equival

ANSWER:

Step 1 of 3

a) The given relation is  iff  so it follows:

i)  since , the relation is a reflexive relation.

ii) , then , therefore  so , hence the relation is symmetric.

iii)  and , so if a variable  is defined such that  and  this will imply that  so the relation  is transitive.

Therefore the relation  iff  is an equivalence relation since it is reflexive, symmetric and transitive.

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