SOLUTION: Can you help me solving this Q , I have problem with understand it Let A be the set of all ordered pairs of positive integers and R be the relation defined on A where (a, b) R

Question 1009267: Can you help me solving this Q , I have problem with understand it

Let A be the set of all ordered pairs of positive integers and R be the relation defined on A where (a, b) R (c, d) means that b � a = d � c.

** Show that R is an equivalence relation.
** Find [(3, 5)] and [(7, 1)]...
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!
This question belongs to abstract algebra, which does not appear in the given categories.

Question:
Let A be the set of all ordered pairs of positive integers and R be the relation defined on A where (a, b) R (c, d) means that b � a = d � c.
** Show that R is an equivalence relation.
** Find [(3, 5)] and [(7, 1)]...

Solution:
Given (a,b)R(c,d) => b-a=d-c

To show that R is an equivalence relation, we need to show
1. R is reflexive, i.e. (a,b)R(a,b) belongs to R
Since, b-a=b-a, we conclude that R is reflexive.

2. R is symmetric, i.e. (a,b)R(c,d) => (c,d)R(a,b)
Since (a,b)R(c,d) => b-a=d-c => d-c=b-a => (c,d)R(a,b), we conclude that R is symmetric.

3. R is transitive, i.e. (a,b)R(c,d) and (c,d)R(e,f) => (a,b)R(e,f).
Since (a,b)R(c,d)=>b-a=d-c, and (c,d)R(e,f)=> d-c=e-f, which in turn implies
b-a=d-c=e-f => b-a=e-f => (a,b)R(e,f) belongs to R. We conclude that R is transitive.

Since R satisfies all three criteria for equivalence relations, R is an equivalence relation.

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