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This is the text from my book:

To define a statement $A$ so that it is true whenever the statement $B$ is true, we write $$A :\Leftrightarrow B$$ and say '$A$ is true, by definition, if $B$ is true'.

I don't quite understand it. How does it differ from $A := B$ or $A \Leftrightarrow B$? Please provide some examples if you can? Thanks in advance.

Edit:

Just to check whether I have really understood this notation, is it correct to write $(X \subseteq Y) :\Leftrightarrow (\forall x \in X (x \in Y))$?

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    $\begingroup$ This is non-standard notation. I would be wary of using it. $\endgroup$ – almagest Apr 8 '16 at 9:34
  • $\begingroup$ Could be a misprint. I would add: B is true whenever A is true. As it stands, it defines $B\implies A$. $\endgroup$ – Dan Christensen Apr 13 '16 at 15:54
  • $\begingroup$ @DanChristensen The author introduces first the symbol $\Rightarrow$ and then this, so not a misprint... $\endgroup$ – Colescu Apr 14 '16 at 14:33
  • $\begingroup$ How does the author define $\implies$? $\endgroup$ – Dan Christensen Apr 14 '16 at 14:37
  • $\begingroup$ @DanChristensen The author just writes $(A \Rightarrow B) := (\lnot A) \lor B$. $\endgroup$ – Colescu Apr 15 '16 at 4:40
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$A:=B$ means that $A$ is defined as $B$.

$A \Leftrightarrow B$ is a theorem: it means that $A$ and $B$ are defined independently and now you are saying that indeed $A$ holds if and only if $B$ holds. This should be followed by some proof.

$A :\Leftrightarrow B$ is again a definition. It means that this is not something you have to prove (you may need to prove that this is a good definition, but that's another problem).

Examples:

$f(x):= x^3$ : definition. Nothing to prove

$f(x)>0 \Leftrightarrow x>0$ : theorem, you should provide a proof for it (in this case it's trivial)

$n$ is even $:\Leftrightarrow n = 2k$ for some $k \in \mathbb{N}$: definition of being even. Nothing to prove

Edit: yes, your example is correct as that's actually the definition of being a subset

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  • $\begingroup$ Another question still. How does one distinguish between $A :\Leftrightarrow B$ and $A := B$? $\endgroup$ – Colescu Apr 8 '16 at 9:53
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    $\begingroup$ @Yuaxiao $:\iff$ is restricted to statements. If you can read it out loud and answer with "true" or "false", use $:\iff$. Using the first two examples, the first reads "f of x" and the second reads "f of x greater than zero". The second one clearly is answerable, the first one isn't. $\endgroup$ – roman Apr 8 '16 at 10:22
  • $\begingroup$ @Saphrosit Maybe you should note that something like: "We define a relation $>$ on $\mathbb{N}$ via $n>m\iff n|m$" is perfectly fine. $\endgroup$ – roman Apr 8 '16 at 10:25
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One possible reading is that A may be true when B is false, in contrast to "A if and only if B" meaning not only that A is true whenever B is, but that A is false whenever B is.

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  • $\begingroup$ Could you please provide some examples? When can it be that $A$ is true when $B$ is false? $\endgroup$ – Colescu Apr 8 '16 at 9:33

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