# Right and Left arrow notation in proof.

I'm studying vector spaces and I'm reading a proof where the authour uses the symbols

$$(\Rightarrow)$$

and

$$(\Leftarrow)$$

when proving a theorem. He doesn't use them in context, but rather before starting a part of the proof.

How should I read this? Does it have anything to do with symbols and notation in logic?

Proposition 1.10

Let $V$ be a $K$-vector space, and let $S \subseteq V$. Then $S$ is a subspace of $V$ if and only if:

1. $\mathbf{0} \in S$
2. $\mathbf{v},\mathbf{w} \in S\Rightarrow \mathbf{v}+\mathbf{w}\in S$
3. $\lambda \in K,\mathbf{V}\in S\rightarrow \lambda \cdot \mathbf{v}\in S$

Proof

$(\Rightarrow)$ It is immediate [sigh] to verify that if $S$ is a subspace of $V$ then $1$,$2$ and $3$ hold.

$(\Leftarrow)$ $1.$ ensures $S \neq \emptyset$ $2.$ implies that $+$ is an operation in $S$. $3.$ implies that $\cdot$ is an action.

Asociativity and conmutativity are deduced from their validity in $V$, the neutral element from $1.$, and the inverse additive from $3$. The properties of the action are deduced from their validity in $V$.

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He's probably proving a statement of the form "$A$ if and only if $B$", and he's using the first arrow to indicate he's starting the proof of the $A$ implies $B$ part, and the second arrow to indicate he's starting the proof of the $B$ implies $A$ part.

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I added the example. It seems it is the case you're explaining, isn't it? –  Pedro Tamaroff Mar 8 '12 at 5:16
Ha! I just taught that theorem a couple of days ago! Yes, I think my answer fits what you've added. –  Gerry Myerson Mar 8 '12 at 6:20
Great! I accepted. I reached my vote limit, so, I owe you a vote. What does "AB (Harvard)" mean? (I was just peeking your Univ. profile) I guess MS means Master in Science or something of the sort. –  Pedro Tamaroff Mar 8 '12 at 6:23
@PeterT.off Bachelor of Arts, but in Latin (so something like Artis Bacca-something-or-other). –  Gerry Myerson Mar 8 '12 at 6:37
@GerryMyerson It's artium baccalaureus, en.wikipedia.org/wiki/Bachelor_of_Arts –  Doug Spoonwood Mar 9 '12 at 4:03