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

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