abstract algebra help please!

[cuti3panda]

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These questions are relative to Equivalence Relations....

 

Question[1]..Let S be the set of real number. If a,b exist in S, define a~b if a-b is an interger. Show that ~is an equivalence relation on S. Describe the equivalence classes of S.

 

Question[2]... Let S be the set of intergers. If a,b exist in S, define aRb if ab>=0. Is R an equivalence relation on S?

 

Question[3].. Let S be the set of interfers. If a,b exist in S, define aRb if a+b is even. Prove that R is an equivalence relation and determine the equivalence classes of S.

 

Hints: 1]..(a,a) exist in R for all a exist in S.. [reflexive property]

2]..(a,b) exist in R implies (b,a) exist in R [symmetric property]

3]..(a,b) exist in R and (b,c) exist in R imply (a,c) exist in R [transitive property]

 

 

thanks a lot

[e(ho0n3]

These questions are pretty straight forward. Just show that each relation is reflexive, symmetric and transitive. If you have any specific problems, let us know.

[MandrakeRoot]

This is straightforward showing the three rules hold.

Let me do one : a ~ a, since a -a = 0 is integer

If a ~ b, then a-b is integer, hence -(a - b) = b-a is integer, thus b ~a

finally if a ~b and b~c , then a - c = (a - b) - (c - b) is integer by the above and the fact that a ~b and b~c , so a ~c.

IT is really easy

 

Mandrake

[cuti3panda]

I can't figure out these problem, if you think it's easy so plzz...help me out with this...thanks a lot

[e(ho0n3]

You need to help us understand what exactly you need help with. Please be specific. Mandrake already did the first question for you. Just follow Mandrake's basic outline to solve the rest of the questions.

[MandrakeRoot]

Say the second question, why dont you arrive to solve it ?

Tell me what you tried to do in order to find the solution and maybe someone can help you find out what is missing in your argumentation ?

 

Mandrake

[matt grime]

Cuti3panda, maths just follows the rules. You know what the rules are that define an equivalence relation, right? So, write them down. Now, for the second one (it isn't an equivalence relation by the way), can you try and find some where where the rules for an equivalence do not hold? It's obviously reflexive and symmetric, so what about transitive? Can you find numbers a,b,c so that ab=>0, bc=>0, but ac<0?

 

the third one just requires you to show the rules for defining an equivlance relation are satisfied. write them out again. and write out what it means for an integer to be even (it is a multiple of two).

[cuti3panda]

Thanks a lot, matt grime

[Dave]

I'd also like to point out that we're not really in the habit of doing people's assignments for them Please don't just post a load of questions and expect the answers straight off; we'll hand out hints, but doing it for you is a bit cheeky and will get you in a lot of trouble if you're caught.

[MandrakeRoot]

I would say that the problem of doing someones exercises for them would be that they learn nothing. The whole point of such exercises (as the one above) is to get comfortable with definitions and simple application of such definitions.

 

Mandrake

[Guest subbiahm]

I support Mandrakeroot.one should try in his own .That too problems of such type after solving 1 or 2 examples u should try to solve

 

subbiah

[Dave]

I would say that the problem of doing someones exercises for them would be that they learn nothing. The whole point of such exercises (as the one above) is to get comfortable with definitions and simple application of such definitions.

 

Mandrake

 

Quite. It is extremely important to get a solid grounding in this stuff, or else you're going to be screwed when you come to the more complex stuff.