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What does the $\Rightarrow$ arrow mean when showing working out in maths?

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How do we use it appropriately?

Brian Fitzpatrick
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confused
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3 Answers3

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The $\Rightarrow$ notation means that if the function on the left hand side of the notation is true, then so is the function on the right hand side of the notation.

So consider $X\Rightarrow Y$. This means that if $X$ is true, then $Y$ is also true.

Brian Fitzpatrick
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lvella
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  • is it necessary/important to include?

    basically we could write that for any equation with ___ = ____

     – confused Mar 14 '14 at 05:26
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    It is necessary to use this sometimes. For example:

    $$ x = 2 => x^2 = 4 $$ In this case we cannot say that they are equal. The => is more of an implication notation to show that logically both sides of the notation are equal.

     – lvella Mar 14 '14 at 05:30
  • Sometimes moving from one equation to the one below it is merely a matter of simplification of one or both sides. In these cases $\implies$ is not helpful (imho). But in other cases, the first equation is used to logically imply a different equation. And in those cases, it adds something by distinguishing the situation from the situation where you are merely simplifying. – 2'5 9'2 Mar 14 '14 at 05:33
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    You can think of it as standing for the word "implies" if you wish. That is how I usually read it in my head. – ricardio Mar 14 '14 at 05:48
  • What is necessary/important to include is: an explanation of what you are doing. Simply writing down some equations may not be enough. – GEdgar Jun 04 '15 at 15:15
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    No, wouldn't that be $X \Leftrightarrow Y \ ?$ and this is where we get confused... – Mr Pie Oct 23 '17 at 05:56
  • Here is an example involving a function, so maybe it demonstrates @alex.jordan 's point about doing "more than simplifying". On page 37 we see that a norm function $f(...)$ takes a vector $x$ and produces a single scalar value. This norm $f$ function must satisfy specific constraints, firstly that $ f(x) = 0 ⇒ x=0$ . Based on other comments here, we can understand this like: "If the norm of a vector is zero, that implies the vector is zero". – Nate Anderson Oct 28 '17 at 15:37
  • what about the single arrow ? I always get confused between the two. -> – moth Jul 22 '21 at 01:20
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    It also means (implicitly yet rigorously) that if $Y$ is false then $X$ is false and if $X$ is false $Y$ maybe false or true. In this sense $\Rightarrow$ is simply a logical connective like and, or, etc. A widespread abuse of notation is to tacitly assume that $X$ is true, which leads in very few cases to logical disasters. I never use it (and discourage my students from) outside the "hypothetical" $X$ case. – Oskar Limka Oct 29 '21 at 17:00
  • A common pitfall with this notation, is to infer (or assume the reader infers) that $X$ be true. – Oskar Limka Jan 08 '22 at 14:49
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It stands for "implies that". For example, $x = 2 \implies x^2 = 4$ - if $x$ is $2$, then it is obvious that $x$ squared is $4$; the symbol essentially shows a function here.

jameselmore
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E. Pereira
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The OP use of $\Rightarrow$s is correct. It is the "let" that is syntactically ambiguous. Are you assuming "$a=2^x$" or are you assuming the series of implications? The point is that mathematically trained people can deal with this abuse of notation in most cases and insisting otherwise in a piece of homework would be seen as pedantic. Writing such passages in a research paper is usually frowned upon in the same way as bad grammar would be frowned upon.

Oskar Limka
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  • The syntactic ambiguity is important, in my view amounting to slightly worse than ‹poor grammar› but possibly actually faulty semantics (failing to convey intended meaning while also plus valid). Reading the OPost as is (without justification), I would take it to be the weaker claim (i.e., the chain of implications, rather than also the claim-as-true unto initial statement $a=2^x$, per safe translation of «Let …»). In light of this, my personal practice is to use single-bar arrows for conditional-truths (longer ones when mixing with double-style, for emphasizing distinction or perhaps . . . – user946772 Dec 20 '21 at 01:58
  • ᠁potential possibility of non-truth, translating the foreward-direction of “$A→B$” as “If $A$[ is true] then $B$[ also is true]”), while reserving short double-bar arrows for assertional truth (translating the foreward-direction of “$P⇒Q$” as “Since $P$ holds, so does $Q$” or “$P→Q; P.:∴ Q$” or “$Q ∵ [P:∧:P→Q]$”. So instead of “Let $A$. $⇒B ⇒C ⇒D ⇒E$”, perhaps re-phrase it slightly as “Let $A$. $⟶B. ⇒C ⇒D ⇒E$”. – user946772 Dec 20 '21 at 02:05