1. Equivalence classes

For the following f: R-> Z is defined by f(x) = floor (x), describe the equivalence classes of the kernal relation of f that partition the domain of f.

Describe the equivalence classes for x~y iff x mod 2 = y mod 2 and x mod 4 = y mod 4 for each of the following relations on N

2. some help

I am not sure I understand the wording of the first question, but if we are thinking of $\displaystyle \mathbb{R}$ and $\displaystyle \mathbb{Z}$ as additive groups, the kernel of f is everything that gets sent to 0 by the floor function, so $\displaystyle ker(f) = [0,1)$.

The other two questions are just giving you the additive cyclic groups $\displaystyle \mathbb{Z}_2, \mathbb{Z}_4$ respectively. Think clock arithmetic.

The first splits every natural number into two sets, the evens- $\displaystyle \bar 0$ and the odds $\displaystyle \bar 1$.

The other splits the natural numbers into 4 distinct equivalence classes based on their remainder when dividing by 4. So by the Euclidean algorithm, you get $\displaystyle \{\bar 0, \bar 1, \bar 2, \bar 3\} = \frac{\mathbb{Z}}{4\mathbb{Z}}\cong \mathbb{Z}_4$

Hope that helps.

3. as this is for a discrete math course, i am not sure that they explicitly deal with abstract algebra. so while your answers seem correct, i would state them differently. the wording of the first has me at a miss also. but i am thinking the problems are to be done something like this...
Originally Posted by Gamma
I am not sure I understand the wording of the first question, but if we are thinking of $\displaystyle \mathbb{R}$ and $\displaystyle \mathbb{Z}$ as additive groups, the kernel of f is everything that gets sent to 0 by the floor function, so $\displaystyle ker(f) = [0,1)$.
probably what they want you to say here is that two numbers are in this equaivalence class iff they are both in the interval you mentioned.

The other two questions are just giving you the additive cyclic groups $\displaystyle \mathbb{Z}_2, \mathbb{Z}_4$ respectively. Think clock arithmetic.

The first splits every natural number into two sets, the evens- $\displaystyle \bar 0$ and the odds $\displaystyle \bar 1$.
here we would say "under this relation, two numbers are in the same equivalence class if they have the same parity." that is, this relation partitions the natural numbers into odd and even numbers. which is what you said, but i think it should have been stated differently.

The other splits the natural numbers into 4 distinct equivalence classes based on their remainder when dividing by 4. So by the Euclidean algorithm, you get $\displaystyle \{\bar 0, \bar 1, \bar 2, \bar 3\} = \frac{\mathbb{Z}}{4\mathbb{Z}}\cong \mathbb{Z}_4$

Hope that helps.
here we would say, "under this relation, two numbers are in the same equivalence class if they have the same remainder when divided by 4." rather than go into the whole thing about factor sets and cyclic groups.

i guess cj can pick which answer he's looking for