Need Some Math Help

[Anubis]

Ok I try not to ask you guys help on work and stuff but im Stuck on this Descrete math Problim
Any help at all would be good


Let SR^2)-->R be the Sum function S(x,y)=x+y. Where R is the set of Real numbers

Prove or Dissporve
a) S is Injective (One to One)
b) S is Surjective (Onto)

 

[DeafeningSilence]

Sorry... I'm useless.

 

[FeathersMcGraw]

Originally posted by: DeafeningSilence
Injective? Yes. Two inputs have only one possible output.

Surjective? No. Any given output can be the result of infinite inputs. (ex: 5 = 2+3 or 1+4 or .12+4.88, etc.)

I do not believe that you are remembering the proper definitions of injective and surjective functions.

 

[Anubis]

well i know the defs. can anyone provide a possible solution

 

[Hanpan]

I will take a shot.


a) disprove by counterexample. 2+5=7 6+1 =7

b) this one is a bit more tricky to prove.


Let z be an element of R

if s to not to be surjective then there must excist a z such that x+y/=z (xeR,yeR)
since there need be only one zeR for this to be true we can fix y = 0
we now have there excist an xeR such that x/=z
we now replace x with w which is defined as z+1 (zeR therefore z+1 eR if you teacher is a stickler you can prove this as well)
we now have z=/z+1 which is a contradiction

Therefore there does not excist an zeR such that x+y/=z. This shows the function is surjective.

In words.

A function is one to one if for every element of set b that is mapped there is only a single corresponding element of set a. This is clearly not the case.

A function is onto if every element of set b is mapped by a. This is easily shown that for every element in b one can take that element subract one and then add an element of a (namely one) and have the same number.
So the function is onto.

Please note it has been a while since I did discreet math so my proofs may not be of the rigour demanded by your prof.

If there are any mistaked in the above analysis please point them out and i will be happy to correct myself.

 

[Simoncifer]

Yes, finally a math question that is close to Algebra.....

To TheEvil1:

I think DeafeningSilence has the definitions of injective and surjective cross-wired.

Anyway, onto answering your question.....


for injectiveness, the answer would be "No", as shown in the example by DeafeningSilence,
S(1,4) = 5 = S(2,3)


for surjectiveness, well the easiest way I could think of is that by showing that:

Let take a number from R and call it y

y-1 would belong to R also. Now, since
S(y-1, 1) = (y-1) + 1 = y
we find a member in R^2 that would correspond to y by S, namely, the pair (y-1, 1)

I think you can draw the conclusion from there on .....