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I'm reading a paper that's trying to prove a theorem. However there is a weird notation that I couldn't understand.

First they present the theorem and then they present two claims. In the first claim it has an arrow that directs to the right, and then a second claim with an arrow that directs to the left. I couldn't understand the meaning of this notation. What does it mean?

Actually I couldn't understand how the two claims prove the theorem and I guess the secret is the the notation. Because the first claim is so strange to be introduced since it's against the goal of the theorem!

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  • $\begingroup$ You prove a theorem. To do so, you provide a proof. $\endgroup$
    – Pedro ♦
    Commented Oct 15, 2014 at 18:50
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    $\begingroup$ The theorem states an "if and only if", $A \iff B$. The first part proves the implication $A \Rightarrow B$, the second the implication $B\Rightarrow A$, or $A \Leftarrow B$. $\endgroup$
    – Daniel Fischer
    Commented Oct 15, 2014 at 18:50

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The statement that is being proved is a biconditional statement. The key phrase in the theorem is "if and only if". The biconditional statement $A$ if and only if $B$ can be written symbolically as $A\Longleftrightarrow B$.

You prove a biconditional statement by proving $A\Longrightarrow B$ and $A\Longleftarrow B$. That is what the symbols are trying to show you. The first paragraph proves $A\Longrightarrow B$ and the second paragraph proves $A\Longleftarrow B$.

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