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There are two methods for attempting to disentangle this confusion that are generally tried, the first more rarely, the second quite frequently, though apparently in opposite proportion to their respective chances of actual success.  In order to describe the rationales of these methods I need to introduce a number of technical concepts.

There are two methods for attempting to disentangle this confusion that are generally tried, the first more rarely, the second quite frequently, though apparently in opposite proportion to their respective chances of actual success.  In order to describe the rationales of these methods I need to introduce a number of technical concepts.

Suppose <math>P\!</math> and <math>Q\!</math> are dyadic relations, with <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z.\!</math>  Then the ''contension'' of <math>P\!</math> and <math>Q\!</math> is a triadic relation <math>R \subseteq X \times Y \times Z\!</math> that is notated as <math>R = P\!\!\And\!\!Q\!</math> and defined as follows.

Suppose <math>P\!</math> and <math>Q\!</math> are dyadic relations, with <math>P \subseteq X \times Y\!</math> and <math>Q \subseteq Y \times Z.\!</math>  Then the ''contension'' of <math>P\!</math> and <math>Q\!</math> is a triadic relation <math>R \subseteq X \times Y \times Z\!</math> that is notated as <math>R = P\!\!\And\!\!Q</math> and defined as follows.

{| align="center" cellspacing="8" width="90%"

{| align="center" cellspacing="8" width="90%"

| <math>P\!\!\And\!\!Q\ ~=~ \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P ~\text{and}~ (y, z) \in Q \}.\!</math>

| <math>P\!\!\And\!\!Q ~=~ \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P ~\text{and}~ (y, z) \in Q \}.</math>

|}

|}

In other words, <math>P\!\!\And\!\!Q</math> is the intersection of the ''inverse projections'' <math>P' = \operatorname{Pr}_{12}^{-1}(P)\!</math> and <math>Q' = \operatorname{Pr}_{23}^{-1}(Q),\!</math> which are defined as follows:

In other words, P&Q is the intersection of the "inverse projections" P' = Pr12 1(P) and Q' = Pr23 1(Q), which are defined as follows:

Pr12 1(P) = PxZ  = {<x, y, z> C XxYxZ : <x, y> C P}.

Pr23 1(Q) = XxQ  = {<x, y, z> C XxYxZ : <y, z> C Q}.

\operatorname{Pr}_{12}^{-1}(P) & = & P \times Z & = & \{ (x, y, z) \in X \times Y \times Z : (x, y) \in P \}.

\operatorname{Pr}_{23}^{-1}(Q) & = & X \times Q & = & \{ (x, y, z) \in X \times Y \times Z : (y, z) \in Q \}.

Inverse projections are often referred to as "extensions", in spite of the conflict this creates with the "extensions" of terms, concepts, and sets.

Inverse projections are often referred to as "extensions", in spite of the conflict this creates with the "extensions" of terms, concepts, and sets.