The equation can be verified by establishing the corresponding equation in matrices.
If and are two 1-dimensional matrices over the same index set then if and only if for every Thus, a routine way to check the validity of is to check whether the following equation holds for arbitrary
Taking both ends toward the middle, we proceed as follows.
The products commute, so the equation holds. In essence, the matrix identity turns on the fact that the law of exponents in ordinary arithmetic holds when the values are restricted to the boolean domain Interpreted as a logical statement, the law of exponents amounts to a theorem of propositional calculus otherwise expressed in the following ways.
- Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References
- Logic Syllabus • Relational Concepts • Relation Theory • Relative Term