Discrete Math: Identity

[wessleym]

Here's where I'm stuck:
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Which of the following operations f: Z x Z -> Z have an identity?
a. f(x,y) = x + y -xy
b. f(x,y) = max{x,y}
c. f(x,y) = x^y
d. f(x,y) = x +y - 3
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I would think they would all have the operations "plus zero" and "times one," but I don't think this problem could be that easy. Thanks for the help!

 

[daon]

The identities here are not necessarily the same as those from your integer axioms. You are trying to find a number e in Z such that for all x in Z f(x,e) = f(e,x) = x.

So for a), let x be an elt in Z. f(x,0) = x+0-x*0 = x. And f(0,x) = 0+x-0*x=x. Thus 0 is an identity elt.

For b), Can you find an e such that for all x max{e,x} = x and max{x,e}=x? Think about your domain (ZxZ).

For c), is there an e in Z s.t. for all x in Z x^e = x and e^x=x?

For d) is there an e in Z such that for all x in Z: x+e-3 = x and e+x-3=x?

-Daon

 

[wessleym]

Just to make sure I understand:
b. e=x
c. e does not exist
d. e=3
Thanks a ton for your help. If my answers are correct, please let me know. Thank you!
Edit: And just out curiosity, why does the problem say
"Z x Z -> Z"
?

 

[pka]

There is no identity for part b.
It is correct that there is no identity for c. However 1 is a right-handed identity for c. But to be a proper identity we need it to work on both right and left.

“And just out curiosity, why does the problem say Z x Z -> Z”
Z is the traditional letter that mathematicians use for the set of integers; its use comes from the German word ‘zahlen’ for number. If it were the N for natural numbers then part b would have an identity. The reason they use ZxZ is that all binary operations are defined for a set of ordered pairs. So the functions maps pairs to a element of the set.