Define the relation R on N*N by: (x,y)R(z,w) if and only if x-z = w-y. Check whether R is an equivalence relation. Explain your answer
My teacher answer is:
Using the shortcut method: (a,b)R(c,d) if and only if f(a,b)=(c,d) where f(a,b) = a+b
Here also:
Define the relation R on N, by mRn if 3|m-n (a) Is R an equivalence relation? If so, what are its equivalence classes?
My teacher answer is:
Using the shortcut method: mRn if and only if f(m)=f(n) where f(m) = m % 3
I didn't understand how she proved it this way! I usually prove it by proving if it's reflexive symmetric and transitive.
In the case of the second question all your teacher has done is to show that two pairs are related on R iff they are congruent to each other modulo 3. I am guessing the teacher had previously taught you that congruence modulo n is an equivalence relation on the set of natural numbers.
I am also willing to guess that maybe she taught you that $(x,y)R(a,b) \iff x + y = a +b$ is an equivalence relation. So by the shortcut method I think she simply means transforming the relation into something she had already taught was an equivalence relation.
But if I were you I would just prove they are symmetric, reflexive and transitive. It is not that difficult and would look much better.