Sometimes you will see theorems of the form "Let $H_1, \dots, H_n$. If $A$, then $B$". Sometimes "suppose" or "if" is used instead of "let". Here's an example:
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Let $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.
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Suppose $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.
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If $x\in\mathbb{R}$ and $x\geq 0$, then $|x|=x$.
I'm under the impression that these are all equivalent ways of saying the same thing.
In this example, I would call "$x\in\mathbb{R}$" a hypothesis and "$x\geq 0$" the antecedent.
But in the third statement, is there an unambiguous contrapositive?
In certain contexts, I think it is understood that we're not really considering the case when $x\notin\mathbb{R}$. But "$x\in\mathbb{R}$" is nevertheless part of the antecedent in statement (3). So if we agree (1-3) are equivalent, then I see two contrapositives:
a) If $x\in\mathbb{R}$ and $|x|\neq x$, then $x<0$.
b) If $|x|\neq x$, then $x<0$ or $x\notin\mathbb{R}$.
I think Halmos' Naive Set Theory is an example where form (3) is preferred to (1,2).
The questions are:
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Are those statements equivalent?
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In the third statement, what is The contrapositive? EDIT: Generally, if you see a theorem of the form "Let $H_1, \dots, H_n$. If $A$, then $B$", what is its contrapositive? How do you know?
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Do mathematicians make any effort to separate the hypotheses ($H_1,\dots,H_n$) from the antecedent ($A$) of the claim? If so, how? Or is this one of those things everybody understands and no one is explicit about?
Best Answer
I would say that all three forms are equivalent. They all express your assumptions followed by the consequence of those assumptions. The third one would be more easily translated into a purely symbolic representation, but it's worth noting that mathematics was a rhetorical art long before symbolism. With that in mind, they all state the same idea, that a certain property follows inevitably from certain other properties, and the rest is just stylistic.
You can see some guides for the difference between "suppose" and "let" here. It boils down to "Let $x$ be something" means you're telling me that you're using $x$ as a shorthand for an object with a certain property, while "suppose" can be used for the same thing and also for pretending something is true, such as "Suppose this theorem is true, here are the consequences". It wouldn't make much sense to say, "Let this theorem be true," because you can't just declare a theorem to be true the same way you can claim your arbitrary label $x$ represents a object with a certain property.
It's all very subtle, I hadn't thought about it before with absolute rigor. I doubt most people do. The difference is almost colloquial to math writing, a linguistic quirk rather than a mathematical one.
About finding the contrapositive, all the statements are the same, so we can choose one arbitrarily. I'll pick the third formulation for clarity. If we start with "If $H$ and $A$, then $B$", the contrapositive is clearly "If $\neg B$, then $\neg H$ or $\neg A$", with the antecedent being flipped around by DeMorgan's law. Also note that I combined all the $H_i$'s into one $H$ since it amounts to the same thing.
Now we're trying to prove an OR statement. You could prove it by cases, but there's another way. The statement "If $P$, then $Q$" is equivalent to the statement "$\neg P$ or $Q$". So likewise, if we have "$P$ or $Q$", we equivalently prove "If $\neg P$, then $Q$". It's like proving the OR statement by saying, "Ok, first assume $P$ is true. So the OR statement is true and we're done. Now assume $P$ is not true. We have to show $Q$ is true or else the OR statement isn't always true."
Applying it back to our example, we can prove "If $H$, then $\neg A$" instead of the OR statement in the consequent. So now we're saying, "If $\neg B$, then if $H$, then $\neg$A". That's the same as "If $\neg B$ and $H$, then $\neg A$", just use the earlier trick to rearrange it. So now you can rephrase it as "Suppose $H$. If $\neg B$ then $\neg A$" since those formulations are the same.
So you can keep it in the same domain, it all amounts to the same thing. It all depends on what you want to emphasize.