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I am trying to understand induction on two integer variables Induction on two integer variables

Let's take this case:

  • $P(0,0)$
  • $\forall x,y. P(x,y) \Rightarrow P(x+1,y)$

  • $\forall x,y. P(x,y) \Rightarrow P(x,y+1)$

  • $\forall x,y. P(x,y)$

I don't understand why line 3 is necessary. Line 2 proves the statement $F(y)$ where $F(y) = \{ \forall x, P(x,y) \} $ for arbitrary values of $y$. Why isn't this enough?

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usual me
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2 Answers2

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Suppose that $P(x,y)$ means: I can get to the point $\langle x,y\rangle$ in the plane. The first line then says that I can get to the origin, the second line says that if I’m somewhere in the plane, I can step one unit to the east, and the third line says that if I’m somewhere in the plane, I can step one unit to the north. Given all three of those statements, it’s intuitively clear that I can reach any point of the integer lattice in the first quadrant. (The integer lattice is the set of points whose coordinates are integers.)

Suppose that I had only the first two statements: then I could still reach the origin, which at least gets me onto the integer lattice, but I could only travel east, so I could only reach the points of the lattice that are on the non-negative $x$-axis. Similarly, if I had only the first and third statements, I could reach only the integer lattice points on the non-negative $y$-axis. And if I had only the second and third statements, I couldn’t get onto the integer lattice at all, so the ability to move east and north wouldn’t do me any good.

Thus, in order to be able to conclude that I can reach every point in the non-negative integer lattice, I need all three of the statements.

Brian M. Scott
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I will give a specific example to show line 2 is not enough. Suppose we want to prove $P(3, 3)$. By line 2, we can show $P(1, 3) \implies P(3, 3)$. But how do we prove $P(1, 3)$? We cannot prove $P(1, 3)$ from $P(1, 1)$ without line 3. So as you can see, for two variables, you need to be able "move along both directions".

In terms of your explanation, you can prove $F(y)$ given $P(1, y)$ using line 2. But you cannot prove $P(1, y)$ from $P(1, 1)$ without line 3 since you need to be able to "move along $y$" to be able to deduce it from $P(1, 1)$.

Pratyush Sarkar
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