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When proving a biconditional, sometimes the writer says "I'll prove the converse first". I want to say I'll prove left-to-right first. How to say that? I don't want to put a left to right arrow. I prefer words.

Also, how can I say I'll prove the converse of the statement but my strategy is to prove the contrapositive of the converse. Should I just say "I'll prove the contrapositive of the converse"? I'd like to use the standard lingo, but I'm not actually used to reading such statements. I can't remember seeing anyone using these descriptions.

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When proving a biconditional, sometimes the writer says "I'll prove the converse first". I want to say I'll prove left-to-right first. How to say that? I don't want to put a left to right arrow. I prefer words.

Eevee Trainer suggests "forward implication" in their answer, and I think that sounds good: "I'll prove the forward implication first." The opposite, of course, would be "I'll prove the reverse implication first."

An alternative which I think also sounds good is to say "forward direction" instead of "forward implication."

Also, how can I say I'll prove the converse of the statement but my strategy is to prove the contrapositive of the converse. Should I just say "I'll prove the contrapositive of the converse"?

The contrapositive of the converse is called the inverse. Don't call it the contrapositive of the converse, since that sounds redundant (just like if you called it "the inverse of the converse of the converse").

Putting all this together, you might write something like: "First, I'll prove the forward implication. Next, I'll prove the inverse of the forward implication."

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    $\begingroup$ It's nice that there is a word for "inverse", but I had never heard of it. If I wanted to prove that $A$ implies $B$ and I wanted to begin by proving "the contrapositive of the converse", I believe that I would say explicitly "I'll begin by showing that $\neg A$ implies $\neg B$" in words. For instance, when showing that Zorn's Lemma is equivalent to the Axiom of Choice, I'd write "I'll begin by showing that if Zorn's Lemma is false, then the Axiom of Choice must not hold". $\endgroup$ – fonini Feb 15 at 6:10
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When proving a biconditional, sometimes the writer says "I'll prove the converse first". I want to say I'll prove left-to-right first. How to say that? I don't want to put a left to right arrow. I prefer words.

Usually I just see "forward implication" ($\implies$ in symbols) for the "left to right" implication, and "the converse" ($\impliedby$ in symbols) for the "right to left" implication. These seem clear enough, so I doubt anyone would have issue with you using those phrasings.

Also, how can I say I'll prove the converse of the statement but my strategy is to prove the contrapositive of the converse. Should I just say "I'll prove the contrapositive of the converse"?

I don't really have a proper answer for this one, though, but I feel that your phrasing works just as well.

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    $\begingroup$ I'd strongly recommend to enrich "I'll prove the contrapositive of the converse" with follow-up sentences making it clearer. Too often I witnessed authors jumping right into the proof and me wondering what they are actually up to. E.g. continue with "So let us assume that [LHS is not true], from which we now have to prove that [RHS is not true]." $\endgroup$ – ComFreek Feb 15 at 9:03
  • $\begingroup$ Would you enrich "we now prove the inverse of the forward implication"? Do you feel people don't quite know too well that means? Is it better to remind them? $\endgroup$ – user724963 Feb 16 at 2:21

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