Problem 4: Recall that relation from a set X to itself is called an "equivalence relation" if it is reflexive, symmetric

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Problem 4 Recall That Relation From A Set X To Itself Is Called An Equivalence Relation If It Is Reflexive Symmetric 1 (33.81 KiB) Viewed 60 times

Problem 4: Recall that relation from a set X to itself is called an "equivalence relation" if it is reflexive, symmetric, and transitive. A subset A of X is called an "equivalence class" of if for all a1,42 € A we have that a a2, but also that for all a € A and be (X\A), a and b are not equivalent. (a): Define a relation on the integers such that aRb when a - b is even. Prove that this is an equivalence relation. (b): Let A be any set and consider a function f from A to A. Define a relation such that a, Ra₂ when f(a) f(b). Prove that this is an equivalence relation. (c): What are the equivalence classes in the above examples? (d): Is the relation rRy when r-y<2 an equivalence relation? (e): Given an equivalence relation on X, can an element of X be a member of more than one equivalence class? (a): (You definitely don't have to do this one unless you find it fun) Define an equivalence relation on the real numbers such than a is equivalent to b exactly when their difference is a rational number. What is the cardinality of the set of equivalence classes this relation forms? (B): (You don't have to do this one either) Define a relation R on the set of functions from RR such that fRg when f(x) = g(r) at all but countably many values of r. Is this an equivalence relation?