An exercise in my book has the following facts
$A \lor B $
$C \Rightarrow B$
Then later the books says the above are equivalent to $A \lor C$
In another exercise I have the following facts
$A \lor B $
$B \Rightarrow C$
and again the book says the above are equivalent to $A \lor C$
My intuition says that cannot be true in both cases, since if that were true $B \Leftrightarrow C$
Did the book made a typo or I get something wrong?
You're right, that makes no sense.
In fact, neither of those two are equivalent.
To be exact: the conjunction of $A \lor B$ and $C \rightarrow B$ is not equivalent to $A \lor C$, and neither is the conjunction of $A \lor B$ and $B \rightarrow C$.
It is true that $A \lor B$ and $B \rightarrow C$ together imply $A \lor C$.
But $A \lor B$ and $C \rightarrow B$ do not not together imply $A \lor C$.
We can see all this using a truth-table:
\begin{array}{ccc|c|c|c} A&B&C&(A \lor B) \land (C \rightarrow B) & (A \lor B) \land (B \rightarrow C) & A \lor C\\ \hline T&T&T&T&T&T\\ T&T&F&T&F&T\\ T&F&T&F&T&T\\ T&F&F&T&T&T\\ F&T&T&T&T&T\\ F&T&F&T&F&F\\ F&F&T&F&F&T\\ F&F&F&F&F&F\\ \end{array}
Rows $3$, $6$, and $7$ show that $(A \lor B) \land (C \rightarrow B)$ is not equivalent to $A \lor C$, and in fact row $6$ shows that $(A \lor B) \land (C \rightarrow B)$ does not even imply $A \lor C$
Rows $2$ and $7$ show that $(A \lor B) \land (B \rightarrow C)$ is not equivalent to $A \lor C$, but we do see that whenever $(A \lor B) \land (B \rightarrow C)$ is true, $A \lor C$ is true as well, and hence $(A \lor B) \land (B \rightarrow C)$ does imply $A \lor C$