How would you do problem 12? Let S = {1, 2, 3, 4, 5}. Give an example of an equi

Question

How would you do problem 12?

Let S = {1, 2, 3, 4, 5}. Give an example of an equivalence relation R on S that results in four distinct equivalence classes. What are these equivalence classes for your example? Give an example of an equivalent relation R' on S that results in three distinct equivalence classes. What are these equivalence classes for your example? Let S = {1, 2, 3, 4, 5}. Give an example of an equivalence relation R on S having the property that for every 2-element subset A of S, there are two distinct equivalence classes, both of which have a nonempty intersection with A. A relation R on a nonempty set A is defined to be circular if whenever a R b and b R c, then c R a for all a, b, c elementof A. Prove that a relation R on A is an equivalence relation if and only if R is reflexive and circular. A relation R is defined on Z by a R b if 5a - b is even. Prove that R is an equivalence relation. Describe the distinct equivalence classes resulting from R. A relation R is defined on N by a R b if a^2 + b^2 is even.

Explanation / Answer

(a)Let R be a relation defined on Z by a R b if 5a - b is even.

Let a1,a2 Z. Then 5a1-a1=5a2-a2

a1(5-1)=a2(5-1), thus a1=a2 and so is even (since a is an integer); so we have a1 R a2. Hence, R is reflexive.

Let a, b Z and suppose that a R b. Then 5a-b is even. But since (-b) + 5a = 5a +( -b), also even so b R a. So R is symmetric.

Let a, b, c Z and suppose that a R b and b R c. Then 5a - b and 5b - c are both even and so there exist p, q Z so that 5a - b = 4p and 5b - c = 4q. Hence, a R c and so R is transitive. Thus, R is an equivalence relation.

(b) There are 2 equivalent classes

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