On equality of propositions
After the observation that equality of propositions seems to be underused in mathematical reasoning, I discovered that I myself was --perhaps not surprisingly-- only vaguely familiar with all sorts of equalities. This note --which is not deep at all-- is written to remedy that situation. I shall not repeat that ∧ and ∨ are symmetric, associative and mutually distributive; neither shall I repeat de Morgan's laws.
The following expressions (E0.i) are all equal.
With the possible exception of the last two, (the so-called "Laws of Absorption") this is very familiar ground. Also expressions (E1.i) are all equal.
In the following, = has the lowest binding power
Also expressions (E2.i) are all equal; since (E2.0) is symmetric, they occur in pairs.
The last two were
absolutely new for me; their discovery --in a completely different
context-- provided the incentive to write this note.
Transcriber: Kevin Hely.
Last revised on