Consider the sets A = {a1, a2, ..., am}, B = {b1, b2, ..., bn}, and C = {c1, c2, . . . , cp}, where the elements in each set remain fixed in the order given here. Let R1 be a relation from A to B, and let R2 be a relation from B to C. The relation matrix for Ri is M(Ri), where i = 1, 2. The rows and columns of these matrices are indexed by the elements from the appropriate sets A, B, and C according to the orders already prescribed. The matrix for R1 o R2 is the m × p matrix M (R1 o R2), where the elements of A (in the order given) index the rows and the elements of C (also in the order given) index the columns.
Show that for all 1 ≤ i ≤ m and 1 ≤ j ≤ p, the entries in the ith row and jth column of M(R1) ∙ M(R2) and M(R1 o R2) are equal. [Hence M(R1) ∙ M(R2) = M(R1 o R2).]