Let R be a relation defined on IN X IN by (m,n) R(p,q) =>m+q=n+p.
(a)Show that R is an equivalence relation on INXIN.
(b)Write down the equivalence classes for the pairs (1,1) and (3,4).
(a)Show that R is an equivalence relation on INXIN.
In this question,they ask me to prove that the relation is equivalence
(equivalence=reflexive,symmetric,transitive)
my solution as following:
reflexive:
(m,n)R (m,n)==> m+n=n+m
example:
(1,1)R(1,1)==>1+1=1+1=2
(3,4)R(3,4)==>3+4=4+3=7
Symmetric:
(m,n)R(n,m)==>m+m =n+n
example:
(3,4) and it's symmetric is (4,3)
3+3=6
4+4=8
in the question ,the question they said that the relation is already symmetric ,what is wrong with my solution ?
transitive:
(m,n)R (p,Q) and (p,q)R(x,y)
so that (m,n)R(x,y)
example :
(1,1)R(0,0)==>1+0=1+0=1
(1,1)R(2,2)==>1+2=1+2=3
so that :
(0,0)R(2,2) ==>0+2=0+2=2
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(b)Write down the equivalence classes for the pairs (1,1) and (3,4).
[(3,4)]={(3,4),(4,3),(2,1)(1,3)}
[(1,1)]={(1,1),(0,0),(2,2)}
I want to ask the following question:
How can I determined the number of elements in subset .
they didn't specify if the set IN contain negative number or not
I think that one else has replied because the question is unclear.
If you do not know what is meant by $\displaystyle IN$, then why do you think we know?
Moreover, you yourself seem not to have a good grasp of the basic concepts here.
For example: to show symmetry if $\displaystyle (a,b)\mathcal{R}(x,y)$ you must show $\displaystyle (x,y)\mathcal{R}(a,b)$.
You seem to be confused on basic points.
Hello Plato,
Thank you Plato for your replying ..
Firstly, I ask about the symbol IN because we always refer to the set by one letter and all pages of the book use one letter such as AXB
Secondly , I understand the Relation concept but this question is unclear for me and if you give me a question about the basics of relation ,I will solve it ..
This is a copy for the question ,if you feel my writing is unclear:
Please believe me, I mean disrespect.
Your reply, the above, simply confirms my thinking.
You do not understand this material.
Why would you simply repeat the question?
If you do not understand the notation, what is the point?
You did not address my point that you seem to completely misunderstand the notation of symmetry. WHY?
You need help far beyond what can be given in this forum.
No, I understand the notion of the symmetry :
for example :
if A={3,5,7}
R={(3,3)(3,5)(5,5)(5,3)(5,7)(7,7)}
I can't say that this relation is symmetric because
(5,7) belongs to R
but (7,5) don't belong to R
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I post again to the question because I feel that my writing is not clear ..
I don't request complete solution , I just need hints and understand the question
and If you don't help me ,I don't will be sad ..
this is my responsibility
============================
Thank you again for everything ..
So I expect the author is Rosen. Thanks.
Your submitted solutions look OK to me. Of course I'm not the important critical reader here.
In (a), the fact that R is defined in terms of another equivalence relation can make life pretty easy when it comes to establishing the properties of R.
Also, you've relied on support (a cancellation law, etc.) throughout. That might have been worthy of mention.
In (b), [(1,1)] is just the diagonal subset of INxIN, and [(3,4)] could be considered an off-diagonal subset.
They're each members of the partition of INxIN induced by R.
I think 'a', 'b' should be 'x', 'y'?
The informal extensions of the sets are OK, provided you're taking 'IN' to denote the set of positive integers.