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

Question 1182916: 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 ⋆ transitive?
Is ⋆ an equivalence relation, a partial order, both, or neither?
Thanks in advance!
Answer by ikleyn(52802) (Show Source): You can put this solution on YOUR website!
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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 ⋆ transitive?
Is ⋆ an equivalence relation, a partial order, both, or neither?
Thanks in advance!
~~~~~~~~~~~~~~~~~~~~~~~

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.

Solved.

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