Let f:a to b be a surjective map of sets prove that the relation a if and only if f(a) = f(b) is an equivalence relation whose e

Question

deff fn [24]

4 years ago

13

Let f:a to b be a surjective map of sets prove that the relation a if and only if f(a) = f(b) is an equivalence relation whose e

quivalence classes are the fibers of f

Mathematics

1 answer:

MrRa [10] · Answer 1 · 2022-02-12T02:48:49+03:00

MrRa [10]4 years ago

6 0

Answer:

Step-by-step explanation:

Denote this equivalence relation by $"\sim"$ (i.e, $a\sim b$ if and only if $f(a)=f(b)$ ), is clear that $a\sim a$ is an equivalence relation since $"="$ is. Now, by definition we have that $[a]=\{b\sim a \mid f(a)=f(b) \}=f^{-1}(a)$ .