Show that it is an equivalence class

Question

1. Let $A\neq \emptyset \neq B$ sets and $f:A\rightarrow B$ be a map. We define the fllowing relation: $$a\sim a' \Leftrightarrow f(a)=f(a')$$

   • Show that $\sim$ is an equivalence relation.

   • Show that $f^{-1}(\{b\})$ an equivalence class in respect of $\sim$ for all $b\in \text{Im}(f)$.

2. We define on $(\mathbb{Z}\setminus \{0\})\times \mathbb{N}$ a relation as follows: $$(a,b)\sim (c,d)\Leftrightarrow ad=bc$$ Show that $\sim$ is an equivalence relation.

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I have done the following for question 2:

i) Reflexiv: $(a, b) \sim (a, b) \Leftrightarrow ab = ab$

ii) Symmetry: $(a, b) \sim (c, d) \Leftrightarrow ad = bc$ and $(c, d) \sim (a, b) \Leftrightarrow cb = da$

then $ad = bc \Rightarrow cb = da$.

iii) Transitiv: $(a, b)\sim (c, d) \Leftrightarrow ad = bc$, $(c, d) \sim (e, f) \Leftrightarrow cf = de$ and $(a, b) \sim (e, f) \Leftrightarrow af = be$

then $((ad = bc) \wedge (cf = de))\Rightarrow af = be$.

Could you give me a hint about the question and especially for the second bullet?

Answer 1

To prove your second bullet point, you must show two things:

1. If $a,a' \in f^{-1}(b)$, then $a \sim a'$.
2. If $a \in f^{-1}(b)$ and $a' \notin f^{-1}(b)$, then $a\not\sim a'$.

Both are straightforward applications of the definition of $\sim$.

What you did in answering question 2 is essentially right. It's just that it is not conveniently spelled.
So reflexivity would be proven as $$(a,b) \sim (a,b) \Leftrightarrow ab=ba,$$ which is true, since the product is commutative in $\mathbb Z$.
For symmetry, if $(a,b) \sim (c,d)$, then, by definition, $ad=bc$; again, by commutativity, it follows that $cb=da$, whence $(c,d)\sim(a,b)$.
To prove transitivity, if $(a,b)\sim(c,d)$ and $(c,d)\sim(e,f)$, then $$ad=bc \quad\text{ and }\quad cf=de$$ and we want to prove that $(a,b)\sim(e,f)$, that is, $af=be$. Now, by cancellation, $af=be$ iff $acf=bce$ iff $ade=bce$ (because $cf=de$) iff $e=e$ (because $ad=bc$), which is true, and therefore, $af=be$ and the relation is transitive.