Please help with #4. Describe the partition for each of the following equivalenc

Question

Please help with #4.

Describe the partition for each of the following equivalence relations. For x, y R, x R y iff x - y Z. For n, m Z, n R m if f n and m have the same tens digit. For x, y R, x R y iff sin x = sin y. For x, y R, x R y iff x2 = y2. Let C = The relation R on C given by x R y iff xy = 1 is an equivalence relation on C. Give the partition of C associated with R. Let C be as in Exercise 4. The relation S on C times C given by (x, y)S(u, v) iff xy = uv is an equivalence relation. Give the partition of C times C associated with S. Describe the equivalence relation on each of the following sets with the given partition. For each a R, let Aa = |(x, y) R times R: y = a - x2|, Sketch a graph of the set A for a = -2, - 1, 0, 1, and 2. Prove that is a partition of R times R. Describe the equivalence relation associated with this partition. List the ordered pairs in the equivalence relation on A =

Explanation / Answer

A partition of C is a set of disjoint subsets of C that has union equal to C. The partition associated with R is such that in each subset that is an element of the partition, x,y are both in the subset iff xRy.

We examine for which elements we have xRy. For which elements do we have xy = +/-1?

By inspection, we can see that if we have (+/-i)(+/-i), the product will be +/-(i2) = +/-(-1) = +/-1. Similarly, if we have (+/-1)(+/-1), the product will be +/-1. So i, -i must be in the same element of our partition as each other, and +1, -1 must be in the same element of the partition as each other.

On the other hand, we have (+/-i)(+/-1) = +/-i, so +i, -i cannot be in the same partition with either +1 or -1.

There is only one partitioning of C into disjoint subsets such that i/-i are in the same subset, 1/-1 are in the same subset, and +/-i are in a different subset from +1/-1:

{1, -1}, {i, -i}

is the partition of C. The sets are disjoint but have union equal to C, so it is a partition. R holds between any two elements in a single element of the partition, and R does not hold between any two elements that come from different elements of the partition. Therefore {1, -1} and {i, -i} are the equivalence classes for R, and {{1, -1}, {i, -i}} is the partition associated with R on C.

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