Understanding implications I have a set of questions where to task is to add the correct implication arrow.
I was able to make sense of most of them, but for two problems, I can't figure out why I was wrong.
Here is what I put:




Statement $S1$
Statement $S2$
I answered
The answer in the book is




$\frac{x}{x+1} = 0$
$x=0$
$S1\iff S2$
$S1 \Rightarrow S2$


$(a,b)$ is a point on the line $y=2x-1$
$b=2a-1$
$S1 \iff S2$
$S1 \Rightarrow S2$




For the first one, I can't see how when $x=0$, $\frac{x}{x+1}$ can equal anything other than $0$, so surely the implication works both ways? I presume I'm missing something obvious.
For the second, the point $(a, b)$ is on line $y = 2x -1$, so $b$ must equal $2a - 1$. Again I presume I've missing something here.
Really would appreciate any help here, I've run out of ideas.
First one:
Sufficiency ($\Rightarrow)$: Suppose $x/(x+1) = 0$. This expression already implies $x\neq -1$. So we can multiply it with $x+1$ to get $x = 0$. (True)
Necessity ($\Leftarrow)$: Suppose $x = 0$. It is a trivial result that $x/(x+1) = 0$. (True)
Therefore, the first one is an N-S condition.
Second one:
In set-builder notation, 'the point is on the line' if and only if
$$
(a,b) \in \{(x,y) \in \mathbb{R}^2~|~y = 2x -1\}
$$
and it is equivalent to say that $b = 2a - 1$ given that $(a,b) \in \mathbb{R}^2$ due to Axiom schema of specification.
That is, the second one is also an N-S condition.
The symbols represent the direction of the implication. It means you can read back and/or forwards substituting by &quot;then&quot;. The one with two arrows is &quot;similar&quot; to the equivalence symbol.
In the first, x = 0 does not imply only that x/(x+1)=0, but much other expressions. The same for the line.
Strictly speaking, the statement has a direction which must be rigidly read.