Relations and functions example 5 and 6 Equivalence classes

Define an equivalence relation.

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is divisible by 2} is an equivalence relation. Write all the equivalence classes of R .

Prove that the relation R on the set NxxN defined by (a ,\ b)R\ (c ,\ d) a+d=b+c for all (a ,\ b),\ (c ,\ d) in NxxN is an equivalence relation. Also, find the equivalence classes [(2, 3)] and [(1, 3)].

The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.

Let A={1,\ 2,\ 3,\ ,\ 9} and R be the relation on AxxA defined by (a ,\ b)R\ (c ,\ d) if a+d=b+c for all (a ,\ b),\ (c ,\ d) in AxxA . Prove that R is an equivalence relation and also obtain the equivalence class [(2, 5)].

Let A={1,\ 2,\ 3,\ ,\ 9} and R be the relation on AxxA defined by (a ,\ b)R\ (c ,\ d) if a+d=b+c for all (a ,\ b),\ (c ,\ d) in AxxA . Prove that R is an equivalence relation and also obtain the equivalence class [(2, 5)].

Prove that the relation R on the set NxxN defined by (a ,\ b)R\ (c ,\ d) iff a+d=b+c for all (a ,\ b),\ (c ,\ d) in NxxN is an equivalence relation. Also, find the equivalence classes [(2, 3)] and [(1, 3)].

Let A={1,2,3,ddot,9} and R be the relation in AxA defined by (a ,b)R(c ,d) if a+d=b+c for (a ,b),(c , d) in AxAdot Prove that R is an equivalence relation. Also obtain the equivalence class [(2,5)].

In order that a relation R defined on a non-empty set A is an equivalence relation, it is sufficient, if R

Let R be the relation in the set Z of integers given by R={(a,b):2 divides a-b}. Show that the relation R transitive ? Write the equivalence class [0].