Changes

We have it within our reach to pick up another way of representing 2-adic relations, namely, the representation as [[logical matrix|logical matrices]], and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in [[linear algebra]].

We have it within our reach to pick up another way of representing 2-adic relations, namely, the representation as [[logical matrix|logical matrices]], and also to grasp the analogy between relational composition and ordinary [[matrix multiplication]] as it appears in [[linear algebra]].

First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition ''G'' ο ''H'' of the 2-adic relations ''G'' and ''H''.

First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition ''G'' ο ''H'' of the 2-adic relations ''G'' and ''H''.

Here is the setup that we had before:

Here is the setup that we had before:

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Let us recall the rule for finding the relational composition of a pair of 2-adic relations.  Given the 2-adic relations ''P'' ⊆ ''X'' × ''Y'', ''Q'' ⊆ ''Y'' × ''Z'', the relational composition of ''P'' and ''Q'', in that order, is written as ''P'' ο ''Q'' or more simply as ''PQ'' and obtained as follows:

Let us recall the rule for finding the relational composition of a pair of 2-adic relations.  Given the 2-adic relations ''P'' ⊆ ''X'' × ''Y'', ''Q'' ⊆ ''Y'' × ''Z'', the relational composition of ''P'' and ''Q'', in that order, is written as ''P'' ο ''Q'' or more simply as ''PQ'' and obtained as follows:

To compute ''PQ'', in general, where ''P'' and ''Q'' are 2-adic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes ''a'':''b'' and ''c'':''d''.

To compute ''PQ'', in general, where ''P'' and ''Q'' are 2-adic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes ''a'':''b'' and ''c'':''d''.