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4Fun
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I'm currently reading the section on relations in Velleman's "How to prove it" and I have found a statement somewhere that I want to prove, but I'm not sure whether what I have come up with is reasonable and I also have some questions on the logic used in these type of proofs.
The theorem is: Let R be a relation on set A. Then it holds that:
R^2 \subseteq R <=> R is transitive <=> R^i \subseteq R ,\forall i \geq 1
I'll start with R^2 \subseteq R <=> R is transitive:
=> Assume that R^2 \subseteq R. R^2 means that for an arbitrary ordered pair (x, z)\inA there is a y\inA such that xRy and yRz. The goal is to show that R is transitive, which is basically a conditional statement. If xRy and yRz hold, then we can infer that xRz. Therefore I can assume xRy and yRz, and now should prove that xRz.But because R^2\subseteqR, I know that xRy and yRz is true for an arbitrary (x, z)\inA. It follows that (x, z)\inR.
<= Assume that R is transitive. Therefore if for any (x, z)\inA it is true that there is a y\inA such that xRy and yRz, I know that xRz. Now since R^2\subseteqR is also a conditional statement of the form \forall(x,y)\niAxA((x,y)\niR^2 then (x,y)\niR, I can assume that there is an ordered pair, call it (x,z)\niR^2. Since we assumed that R is transitive, it follows by modus ponens that (x,z)\niR.
Now I'll try to prove: R is transitive <=> R^i \subseteqR, \foralli\geq1.
=> Assume R is transitive. To prove that R^i\subseteqR, for all i \geq1 we can use induction.
Base case: i=1 -> R\subseteqR obviously holds.
Now I'm not really sure whether I actually have to use induction here. I think what needs to be shown here is pretty straightforward. If there is a y\inA such that x1Ry and yRx2 hold, then because R is transitive it follows that x1Rx2. And then you can continue this way with x2 and x3 and in general any value i. But I don't really know how to write this in a formally correct way. Can anybody help me with that? And please give some feedback on the first part of the proof.
The theorem is: Let R be a relation on set A. Then it holds that:
R^2 \subseteq R <=> R is transitive <=> R^i \subseteq R ,\forall i \geq 1
I'll start with R^2 \subseteq R <=> R is transitive:
=> Assume that R^2 \subseteq R. R^2 means that for an arbitrary ordered pair (x, z)\inA there is a y\inA such that xRy and yRz. The goal is to show that R is transitive, which is basically a conditional statement. If xRy and yRz hold, then we can infer that xRz. Therefore I can assume xRy and yRz, and now should prove that xRz.But because R^2\subseteqR, I know that xRy and yRz is true for an arbitrary (x, z)\inA. It follows that (x, z)\inR.
<= Assume that R is transitive. Therefore if for any (x, z)\inA it is true that there is a y\inA such that xRy and yRz, I know that xRz. Now since R^2\subseteqR is also a conditional statement of the form \forall(x,y)\niAxA((x,y)\niR^2 then (x,y)\niR, I can assume that there is an ordered pair, call it (x,z)\niR^2. Since we assumed that R is transitive, it follows by modus ponens that (x,z)\niR.
Now I'll try to prove: R is transitive <=> R^i \subseteqR, \foralli\geq1.
=> Assume R is transitive. To prove that R^i\subseteqR, for all i \geq1 we can use induction.
Base case: i=1 -> R\subseteqR obviously holds.
Now I'm not really sure whether I actually have to use induction here. I think what needs to be shown here is pretty straightforward. If there is a y\inA such that x1Ry and yRx2 hold, then because R is transitive it follows that x1Rx2. And then you can continue this way with x2 and x3 and in general any value i. But I don't really know how to write this in a formally correct way. Can anybody help me with that? And please give some feedback on the first part of the proof.