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I was reading this answer and the person provided two sets of principles.

For the both sets, I understand the base case. In the inductive step for the first set, do I have to prove $$\forall x,y. P(x,y) \Rightarrow P(x+1,y) \wedge P(x,y+1), \text{where $x, y$ are arbitrary}?$$ For the second set of principles, I don't quite get what the inductive step is. I think I am to do a proof by cases, but I don't see why the cases were broken up into $$\forall x,y. P(x,0) \Rightarrow P(x+1,0) \text{ and } \forall x,y. P(x+1,y) \Rightarrow P(x,y+1).$$ What is the logic behind this?

Also what is the difference between these two methods, and when do I use which?

Cheers!

tash
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1 Answers1

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For the first question, yes, together with $P(0,0)$.

For the second question, try to draw a grid to visualize - as a comment on the reply points out, the combination of both steps covers the entire discrete positive plane, and thus the statement is then valid for all natural $x$ and $y$.

Regarding the third question, I can't help - I'd assume there's some sort of problem where this form is more convenient to prove.

Desiato
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