MHB Is this transitive closure correct or not?

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The discussion centers on understanding the representation of a transitive closure using an adjacency matrix. Participants seek clarification on the values in the matrix, specifically why they vary from 0 to 2 in the context of direct transitive closure. The transitive closure is defined as a relation that can be represented by its own adjacency matrix, but confusion arises regarding its representation in a column format. The conversation emphasizes the need for a clearer explanation of the relationship between the adjacency matrix and the transitive closure values. Overall, the thread highlights a common challenge in grasping the concept of transitive closure in relation to adjacency matrices.
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Could you explain what you mean by the table and especially by the column next to it?
 
It is an Adjacency matrix, on the right side is a direct transitive closure. Explain to me why the first one is 0, second - 1, third - 2 etc. I don't understand this transitive closure.
 
Natalie said:
It is an Adjacency matrix, on the right side is a direct transitive closure.
Transitive closure of a relation is another relation. It can be represented by its own adjacency matrix, but I don't understand how it is represented by a column.
 

Thread 'Formal derivation of statement from Peano Arithmetic system'

I was reading a Bachelor thesis on Peano Arithmetic (PA). PA has the following axioms (not including the induction schema): $$\begin{align} & (A1) ~~~~ \forall x \neg (x + 1 = 0) \nonumber \\ & (A2) ~~~~ \forall xy (x + 1 =y + 1 \to x = y) \nonumber \\ & (A3) ~~~~ \forall x (x + 0 = x) \nonumber \\ & (A4) ~~~~ \forall xy (x + (y +1) = (x + y ) + 1) \nonumber \\ & (A5) ~~~~ \forall x (x \cdot 0 = 0) \nonumber \\ & (A6) ~~~~ \forall xy (x \cdot (y + 1) = (x \cdot y) + x) \nonumber...
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