SOLUTION: Determine if the following are equivalence relations: 1. S=ℝ define (a,b)∈R if a=b+1

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Question 1191032: Determine if the following are equivalence relations:
1. S=ℝ
define
(a,b)∈R if a=b+1
Answer by ikleyn(52800) About Me  (Show Source):
You can put this solution on YOUR website!
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Determine if the following are equivalence relations:
1. S=ℝ
define
(a,b)∈R if a=b+1
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A relation R is called "equivalence",  if for any pair (a,b) belonging to R,  the inverted pair (b,a) also belongs to R.


Consider the pair (a,b) = (2,1).               We have 2 = 1 + 1, so the pair (2,1) belongs to R.


Now consider the inverted pair (b,a) = (1,2).  We have  1 =/= 2+1, therefore the pair (b,a) does not belong to R.


It shows and proves that R is not an equivalence relation.

Solved, answered and explained by presenting a counter-example.