Equivalence relation on set of ordered pairs
Let S be the set of ordered pairs of positive integers, let z = (5,8), and define R so that (x1, x2) R (y1, y2) means that x1 + y2 = y1 + x2.
Show that the given relation R is an equivalence relation on the set S. Then describe the equivalence class containing the given element z of S, and determine the number of distinct equivalence classes of R.
A detailed proof of the fact that the given relation R on the set S of all ordered pairs of positive integers is an equivalence relation is provided. The equivalence class containing the given element z is described. A detailed determination of the number of distinct equivalence classes of R is given, as are examples of elements of several different equivalence classes.
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