PlanetMath: therefore sign

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therefore sign

(Definition)

The therefore sign “ $% latex2html id marker 144 $ \therefore$$ ” is used especially in handwritten mathematical text as a shorthand of the word `therefore' or `thus', between some sentences or relations:

$% latex2html id marker 146 $\displaystyle S_1 \quad \therefore \; S_2$$

It expresses that

has been inferred from

or from

and some preceding facts. The sign is rather a punctuation mark than a symbol of logical implication. Grammatically, it could be characterised a conclusive coordinating conjunction. The usage of the symbol is not mathematically well-defined, and it often means `we can conclude in context' or `we can conclude from statements already shown or assumed to be true'.

For example, in determining an angle of a right triangle, one may write

$% latex2html id marker 154 $\displaystyle \sin\alpha = \frac{1}{2} \quad \therefore \; \alpha = 30^\circ$$

Here, “ $% latex2html id marker 156 $ \therefore$$ ” does not represent a proper implication “ $\Rightarrow$ ”, since the exact implication here would be

$\displaystyle \sin\alpha = \frac{1}{2} \;\, \Leftarrow \;\, \alpha = 30^\circ.$

To obtain a strict implication, we would need to introduce some of the context. For instance, we know that, since $\alpha$ is an angle of a right triangle, $0^\circ \le \alpha \le 90^\circ$ , so what we wrote could be interpreted as the implication

$\displaystyle \sin\alpha = \frac{1}{2} \; \land \; 0^\circ \le \alpha \le 90^\circ \;\, \Rightarrow \;\, \alpha = 30^\circ.$

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"therefore sign" is owned by pahio. [ full author list (2) ]

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See Also: rings of rational numbers

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Cross-references: strict, implication, right triangle, angle, well-defined, logical implication, relations
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This is version 5 of therefore sign, born on 2008-03-20, modified 2008-03-29.
Object id is 10425, canonical name is ThereforeSign.
Accessed 462 times total.

Classification:

AMS MSC:	00A05 (General :: General and miscellaneous specific topics :: General mathematics)
	00A06 (General :: General and miscellaneous specific topics :: Mathematics for nonmathematicians )

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