I'm working through a book on my own which has just introduced the ideas of $A \Rightarrow B, B \Leftarrow A$ and $A \Leftrightarrow B$. It then gave 20 exercise questions to answer. I've correctly answered all of them except for two, and I don't understand why my answers are incorrect. The two questions are as follows:
Insert the symbol $ \Rightarrow $, $ \Leftarrow $ or $ \Leftrightarrow $ which fully represents the link between the two statements.
Question 1:
Statement A: $\frac{x}{x + 1} = 0$
Statement B: $x = 0$
My answer: $A \Leftrightarrow B$
Book's answer: $A \Rightarrow B$
For this one, it seems simple enough to see that $A \Rightarrow B$, but I don't understand why $A \Leftarrow B$ is false. It seems no different to expressing $x = 0$ as something like $x + 1 - 1 = 0$. Wouldn't $x = 0$ imply $x + 1 - 1 = 0$?
Question 2:
Statement A: $(a, b)$ is a point on the line $y = 2x-1$
Statement B: $b = 2a - 1$
My answer: $A \Leftrightarrow B$
Book's answer: $A \Rightarrow B$
Similar situation with this one. If $(a, b)$ is a point on the line $y = 2x - 1$, that should imply $b = 2a - 1$. Then surely, if $b = 2a - 1$, $(a, b)$ is going to be a point on the line $y = 2x - 1$. So, again, I don't understand why $A \Leftarrow B$ is false.
Is my logic flawed?