Answer to Question #23581 in Abstract Algebra for Paras

Question

A relation R is defined on the set of integers as xRy iff (x+y) is even. which of the following statement is true ?
a) R is not an equivalence relation
b) R is an equivalence relation having one equivalence class
c) R is an equivalence relation having two equivalence class
d) R is an equivalence relation having three equivalence class

Answer 1

a) is false, since xRy meansthat x,y are either both odd either both even. So, xRx as x+x=2x - even,
xRy and yRx as x+y=y+x
xRy and yRz implies xRz as in this case all x,y,z are either odd either even.
b) false
c) true . All integers can be divided into 2 groups: even numbers and odd. Theninside any these groups all their elements are equivalent.
d) false