We learn the meaning of a word either by seeing it defined in terms of words already known to us or by experiencing examples of its use and sensing the common properties of the examples. For instance, consider these statements: “Snerd is a member of the country club,” “This saucer belongs to my dinnerware set,” “Fromage is one of the French nouns,” “Points A and B are on the perpendicular bisector of angle α,” “The assault on the western flank was a key element of Varnish’s battle plan.” In each of these assertions something is said to be a member or element of some collection or set of things. As the examples suggest, sets are of wildly different sorts and, accordingly, “membership” can have a variety of meanings. (Dictionaries provide no substantial help: a standard dictionary defines a set as a “collection of objects,” a collection as an “aggregate,” and an aggregate as a “collection”; and a member is “one of the elements of which an aggregate is composed.” Check your dictionary.) But mathematics has to start somewhere, so we agree to accept the words set and member as undefined (or primitive) terms, though we take comfort from the meaningful interpretations we give the terms in our daily lives. For linguistic variety we sometimes use the words family and collection as synonyms for set, we use the word element for a member of a set, and we say that an element belongs to any set of which it is a member.
KeywordsEquivalence Relation Prime Number Pairwise Disjoint Venn Diagram Mathematical Induction
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