For each of the following prove that the relation is an equivalence relation. Then give information about the equivalence classes as specified.
1.) The relation R on R(Real Numbers) given by xRy iff x-y is an element of Q. Give the equivalence class of 0; of 1, of sqrt(2).
1.) The relation R on R(Real Numbers) given by xRy iff x-y is an element of Q. Give the equivalence class of 0; of 1, of sqrt(2).
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Reflexive: xRx since x-x = 0, which is rational
Symmetric: If xRy then x-y is rational. But, x-y = y-x so yRx.
Transitive: If xRy and yRz then x-y and y-z are rational. Adding these you get (x-y) + (y-z) = x - z, which is rational, so xRz.
The relation R is reflexive, symmetric and transitive, which makes it an equivalence relation.
For the equivalence classes, proceed as follows:
[0] = Q
[1] = Q
[ sqrt(2) ] = x + sqrt(2), where x is in Q
Symmetric: If xRy then x-y is rational. But, x-y = y-x so yRx.
Transitive: If xRy and yRz then x-y and y-z are rational. Adding these you get (x-y) + (y-z) = x - z, which is rational, so xRz.
The relation R is reflexive, symmetric and transitive, which makes it an equivalence relation.
For the equivalence classes, proceed as follows:
[0] = Q
[1] = Q
[ sqrt(2) ] = x + sqrt(2), where x is in Q