Chapter 9.5, Problem 3E

ch of these relations on the set of all functions fromZtoZare equivalence relations? Determine the properties of an equivalence relation that the others lack.

a) $\left\{\left(f,g\right)|f\left(1\right)=g\left(1\right)\right\}$

b) $\left\{\left(f,g\right)|f\left(0\right)=g\left(0\right)\text{ or }f\left(1\right)=g\left(1\right)\right\}$

c) $\left\{\left(f,g\right)|f\left(x\right)-g\left(x\right)\text{=1 for all }x\in Z\right\}$d) $\left\{\left(f,g\right)|\text{for some C}\in Z,\text{for all }x\in Z,f\left(x\right)-g\left(x\right)\text{=C}\right\}$e) $\left\{\left(f,g\right)|f\left(0\right)=g\left(1\right)\text{ and }f\left(1\right)=g\left(0\right)\right\}$