SOLUTION: A relation ⋆ is defined on z by x ⋆ y if and only if there exists k ϵ z such that y=x+5k . Is ⋆ reflexive? Is ⋆ symmetric? Is ⋆ anti-symmetric? Is ⋆ transiti

As defined in the post, two integer numbers x and y are in relation  " ⋆ "  if and only if their difference  x - y is a multiple of 5.


Therefore, if you are familiar with the definition of the terms, this relation is


    - reflexive :  (x⋆x)  is TRUE for any integer  number x, since  x-x = 0 is a multiple of 5;

    - symmetric :  (x⋆y) implies (y⋆x),  since if x-y is a multiple of 5, it implies that y-x is a multiple of 5;

    - tranzitive : (x⋆y) and (y⋆z) implies (x⋆z), since if x-y is multiple of 5 and y-z is a multiple of 5,

                                                  then  x-z is a multiple of 5, too.


Finally, since the relation  " ⋆ " is reflexive, symmetric and transitive (as we proved it above), it is equivalence relation, by the definition.