What does the $\Rightarrow$ arrow mean when showing working out in maths?
How do we use it appropriately?
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$\begingroup$ It means "then", "hence", "therefore", or any other words in the same spirit. $\endgroup$– TunococMar 14, 2014 at 5:18
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1$\begingroup$ Typically this $\therefore$ means "therefore" and this $\implies$ means "implies". $\endgroup$– Mr PieOct 23, 2017 at 5:56
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3 Answers
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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.
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$\begingroup$ is it necessary/important to include? basically we could write that for any equation with ___ = ____ $\endgroup$– confusedMar 14, 2014 at 5:26
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2$\begingroup$ 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. $\endgroup$– lvellaMar 14, 2014 at 5:30
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1$\begingroup$ You can think of it as standing for the word "implies" if you wish. That is how I usually read it in my head. $\endgroup$– ricardioMar 14, 2014 at 5:48
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1$\begingroup$ No, wouldn't that be $X \Leftrightarrow Y \ ?$ and this is where we get confused... $\endgroup$– Mr PieOct 23, 2017 at 5:56
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1$\begingroup$ 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. $\endgroup$ Oct 29, 2021 at 17:00
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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.
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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.
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$\begingroup$ 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 . . . $\endgroup$– 11qq00Dec 20, 2021 at 1:58
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$\begingroup$ ᠁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 $A$. $⟶B. ⇒C ⇒D ⇒E$”. $\endgroup$– 11qq00Dec 20, 2021 at 2:05