a note on equivalence relations

One purpose of equivalence relations is to maintain premium quality. For example, if we are listing Pythagorean triples, and have included 3-4-5, then including 6-8-10 would be a significant drop in quality, because it takes up time and resources to record it, but it gives no new information, being easily obtainable from the ‘primitive’ Pythagorean triple 3-4-5 by multiplying it through by 2. So, in this case we are interested in only one representative from each such equivalence class. Another example is the specification of the ordering of more than 2 objects. If x, y, and z are three distinct objects, then most of the time we would want to consider the two orderings (x,(y,z)) and ((x,y),z) as equivalent, and write simply ‘(x,y,z)’ to refer to either one.
Such deliberate ignoring of differences is usually called ‘mod-ing out’, and the resulting structure (of equivalence classes) is called a ‘quotient’ structure.
keywords: Mathematics, Logic

lorem ipsum

demographics: the kind of numbers that happen when other numbers are not known.
keywords: Mathematics, calculation, computation, numeracy
(posted to CI on 28.Jun.2019)

the cat’s tail in solving mathematics problems

If you are having difficulty with a mathematics homework (or tournament) problem that asks you to prove something, you might find it very helpful to try to find a numerical example where the stated theorem fails. Your failure to show that the theorem fails can greatly deepen your understanding of what is happening, thereby helping you to come up with the required proof. This is similar to the hunting use that a cat makes of its tail. Its tail serves not only to help it keep, or recover, its balance, but also to flush out prey. For example, when a hungry cat approaches a bush containing a katydid, the katydid’s defense is to remain motionless. In this situation, the cat cannot distinguish the katydid from the foliage, but the cat has a countermeasure to the katydid’s defense: lift its tail up and wag it slowly. The katydid sees the moving tail, and instinctively locks its gaze on it, but this means that its eyes are moving, and that tiny bit of motion is sufficient for the cat to distinguish the katydid.
A related concept is the military maxim of ‘boots on the ground’. That is, it is possible to destroy a country by means of bombardment, but it is possible to control the country only by means of on-site soldiers, that is, ‘boots on the ground’.
keywords: Mathematics, the art of problem solving, solution