Let R be a relation defined on the integers Z by aRb if 3a2

Let R be a relation defined on the integers Z by aRb if 3a^2 - 2b^2 Greaterthanorequalto 0. Which of the properties reflexive, symmetric, and transitive does R possess? Justify your answer! Let R be an equivalence relation on the set S = (a, b, c, d, e). If the distinct equivalence classes of R are {a, c} and {b,d,e}, then what is R? Let R be an equivalence relation on the set T = {1, 2, 3,4,5,6} having the following properties: What are the distinct equivalence classes of R? Justify your answer!

Solution

1. The relation R of equality is defined by a R b on integer Z if 3a^2 - 2 b^2>=0.

Let a and b be specific elements of S.As a b. and b a.

Let be a relation on a set S. Proving is transitive.

2. A leaner definition is: If R is an equivalence relation on a set S= {a,b,c,d.e), then we define the equivalence class of an element x S to be the set of all elements of S equivalent to x.Here equivalence classes of R are (a,c) and (b,d,e) be the set of all elements of S equivalent to R.

3. A relation on the set T = {1, 2, 3, 4, 5, 6} is R = {(1, 1),(2, 2),(3, 3),(4, 4),(5, 5),(6, 6),(1, 3),(1, 6),(6, 1),(6, 3), (3, 1),(3, 6),(2, 4),(4, 2)}.

Thus R is an equivalence relation on T. What are the elements of T that are related to 1? These are {1, 3, 6}. We give this set of elements of T related to 1 a notation: [1] = {x T : x R 1}. We call this set the equivalence class of 1. The equivalence class of a T is [t] = {x T : x R t}. We can identify the equivalence classes of the other elements of T: [2] = {2, 4}, [3] = {1, 3, 6}, [4] = {2, 4}, [5] = {5}, [6] = {1, 3, 6}. Notice that [1] = [3] = [6] and [2] = [4], and so there are only three distinct equivalence classes, and each x T belongs to exactly one of these three. Notice also that the three distinct equivalence classes [1], [2], and [5] are disjoint subsets of T and that their union is all of T.