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Let $G$ be a nonempty set closed under an associative product, which in addition satisfies :

A. There exists an $e$ in G such that $a⋅e=a$ for all $a∈G$.

B. Given $a∈G$, there exists an element $y(a)∈G$ such that $a⋅y(a)=e$.

Prove that $G$ must be a group under this product.

This is a question from I N Herstein. It was already asked before. Here is the solution

https://math.stackexchange.com/a/1200617/581242

Let, $ab=e\land bc=e\tag {1}$ for some $b,c\in G$. And, $ae=a\tag{2}$ From $(2)$, $$eae=ea\implies(ab)a(bc)=ea\implies ((ab)(ab))c=ea\implies ec=ea\tag{3}$$

Similarly, $$ae=a\implies a(bc)=a\implies (ab)c=a\implies ec=a\tag{4}$$

$(3)$ and $(4)$ implies, $$ea=a$$

Also from $(3)$ and $(1)$, $$(bab)(bca)=e\implies b((ab)(bc)a)=e\implies ba=e$$

I don't understand the notation used in first line. What does $$ab=e\land bc=e$$ mean?

José Carlos Santos
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Shweta Aggrawal
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    The $\wedge$ means "and." – saulspatz Feb 07 '19 at 10:58
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    A fun fact, symbol $\land$ is also used for minimum and wedge product. In this case, it's logical "and" though. – Jakobian Feb 07 '19 at 11:07
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    As a tip for the future, you can do a search on the internet for, say, "\land meaning LaTeX" and find the answer quickly. And when you don't know the code for a symbol, you can find it on one of Wikipedia's lists of mathematical symbols or use Detexify to find the TeX code as an alternate starting point. – Mark S. Feb 07 '19 at 11:08
  • @MarkS. Thank you for the advice:) – Shweta Aggrawal Feb 07 '19 at 11:09
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    @Jakobian: in Boolean algebra, and/or are indeed the minimum/maximum operators. There is also a close connection with $\cap/\cup$. –  Feb 07 '19 at 11:14
  • @YvesDaoust In Stochastic processes, we usually write minimum like that. That's what I was relating to. I know that in Boolean algebra $\lor$ and $\land$ operators create an ordering as well, and in that ordering, $\land$ can be interpreted as the minimum. I didn't think about that, but it gives me a whole different view on why is $\land$ used as minimum in stochastic processes. Thank you. – Jakobian Feb 07 '19 at 11:49
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    Bit of advice: Unless you are doing symbolic logic and have to, do not follow that solution's poster by using $\wedge$ and $\vee$ for and/or when writing proofs. Apart from causing the problem you have just illustrated yourself, it is simply unnecessary. Some people get carried away when they get handy with symbols. – rschwieb Feb 07 '19 at 18:11

2 Answers2

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The symbol $\wedge$ means “and”. So, the first line means “$ab=e$ and $bc=e$”.

José Carlos Santos
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In logic, $\land$ means "and" and $\lor$ means "or."

J. W. Tanner
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