4.2 Composite

Two binary relations can form a composite, which is also called a relative product. Let $L_1 \subseteq X_1 \times X_2$ and $L_2 \subseteq X_2 \times X_3$ . Then the composite, called $L_1L_2$ , is defined as follows: $L_1L_2 \subseteq X_1 \times X_3, L_1L_2 = \{(x,z)\vert(\exists(x,y) \in L_1) \wedge (\exists(y,z) \in L_2)\}$

Given a composite relation $L_1L_2$ , then the inverse of the composite is $(L_1L_2)^{-1} = L_2^{-1}L_1^{-1}$ .

Given $L_1$ , $L_2$ and $L_3$ are relations, then $(L_1L_2)L_3 = L_1(L_2L_3)$ .