The sixteen two-input operator CMs

This overview explains how four Boolean cases form an operator correspondence matrix, and why sixteen such matrices are possible. A declared axis order makes every cell interpretable and prevents confusing one case with a whole function. For these operator matrices, use true-first axes. The upper row means A one and the lower row means A zero. The left column means B one and the right column means B zero. And has one only at the upper-left cell, where both inputs are one. A transpose changes a row into a column. Take column one zero and row zero one. Multiply each row entry by each column entry using and: one and zero is zero, one and one is one, zero and zero is zero, and zero and one is zero. The resulting array is zero one, zero zero. This column-times-row construction is called an outer product; a single-cell building piece is also called a basis dyad. Each of the four cells can independently be zero or one. Four binary choices give sixteen possible complete arrays. This counts Boolean functions of two inputs. The four input assignments and the sixteen possible functions are different counts. X labels the rows and Y labels the columns, with true before false on each axis. The four cases are 11, 10, 01, 00. One cell records one case. All four output bits form one operator correspondence matrix. Bracket Theta names the chosen operator matrix. AND returns one only when both inputs are true, giving 1000 in that order. True is represented by the column one, zero; false by zero, one. Transposing a column ket makes a row bra. The outer product of ket one and bra zero places a single one at the top right. The four basis dyads select the four positions. Each of four output positions has two choices. 2⁴ gives sixteen possible two-input functions. Every small grid in this table is a complete function under the same true-first axes. Combining the required basis dyads with exclusive-or builds a matrix: AND uses one, OR three, and XOR the two off-diagonal dyads. Equivalence has ones on the agreement diagonal. For X implies Y, only the case X true and Y false fails. Reversing the operands moves the failing case to the other off-diagonal cell. The axis labels keep that distinction explicit. Work a second complete case with A 0, B 1. Keep this assignment fixed throughout the calculation. Each symbol retains the meaning already introduced; a new value does not create a new logical rule. A zero selects the lower row, while B one selects the left column. The implication matrix contains one at that intersection. Reverse the operand roles: one implies zero is false. Tracking this asymmetric pair exposes a transposition error that a symmetric and matrix could conceal. This case establishes a specific contrast. Operand order changes implication. Axis labels are required. Keep the input, intermediate values, and conclusion together when comparing it with the previous case. The calculated answer belongs to this complete assignment under the stated rule. Or accepts the true-true cell and both mixed cells. Exclusive-or accepts only the two mixed cells. Compare their upper-left entries: or returns one when both inputs are true, while exclusive-or returns zero. That single cell captures the difference between inclusive and exclusive alternatives. Equivalence accepts both equal-input cells, true true and false false. Its matrix has ones on the diagonal in this declared order. The name describes agreement of the two input values; it does not mean every expression containing this operator is equivalent to another expression. Implication fails when A is true and B is false. The reversed input gives one. If we exchange operand roles, we must also exchange row and column roles by transposing the matrix. Symmetric operators can hide an ordering mistake, so implication is a useful check of orientation. Four cases describe one two-input function. Choosing their four outputs gives sixteen possible functions.