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ProofWiki

What does the upside-down A on this page mean?

$$\forall a,b\in S: a\circ x=b\circ x \implies a=b$$

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    $\begingroup$ It means : "for all" $\endgroup$ – Josh Dec 11 '12 at 20:05
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    $\begingroup$ This means for all choices of $a$ and $b$ in $S$. You should read $\forall$ as "for all" $\endgroup$ – Deven Ware Dec 11 '12 at 20:05
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    $\begingroup$ One resource I have found exceedingly useful in determining the meaning of symbols in math texts (especially when I'm teaching myself) is the wiki page on math symbols: en.wikipedia.org/wiki/List_of_mathematical_symbols $\endgroup$ – apnorton Dec 11 '12 at 20:14
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The symbol $\forall$ means "for all." The symbol $\in$ means "in" or is an element of.

In your context, it means for all $a$ and $b$ in $S$.

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$\forall$ means "for all", so the statement $$\forall a,b\in S:a\circ x=b\circ x \Rightarrow a=b$$ means: For all $a$ and $b$ in (elements in) $S$ it is true that if $a\circ x = b\circ x$ then $a =b $.

So saying a bit more: this is just saying that if you have an algebraic structure $(S,\circ)$ like for example a multiplicative group then $x$ is right cancellable if for all $a$ and $b$ if you have that $ax = bx$ then you can automatically conclude that $ a = b$.

Note for example that in the integers minus zero ($\mathbb{Z}\setminus \{0\}$) under multiplication (which actually is not a group) every element satisfies this because you know that if $$a\cdot x = b\cdot x,$$ then the only way that is going to happen is if $a =b$. This is just saying that you can't for example multiply $3$ by anything but $5$ to get $15$.

The reason we can't include $0$ in this example is that for example $$ 4\cdot 0 = 7\cdot 0. $$

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