By David Eugene Smith

A historic account of arithmetic in the USA from the 16th throughout the 19th centuries, with concentrations on schooling, examine, and functional use. moderate shelf put on. prior owner's identify on entrance pastedown. x, 210 pages. cloth.. small 8vo..

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**Example text**

Any given function f defines an equivalence relation in its domain. Equivalent elements are those whose images under f coincide. The equivalence relation generated by a given function / is called the equivalence kernel of / and is denoted by ker (/). 7 illustrates the a ß y Fig. 7. situation on a simple example. In passing we note that if / is injective, then each equivalence class generated by ker(/) consists of a single element of X. Conversely, suppose now that we are given an equivalence relation Ε in some set X.

It includes the possibility of finiteness. " Cardinal Numbers 2 1 3 4 2 2 2 2 2 3 4 3 3 3 3 2 3 4 4 4 4 4 2 3 4 45 Now, for example, let us list elements proceeding via "right to left diagonals": 1 Γ 1 2 2 Γ 1 2 3 3' 2 1 Γ 2 A 3 4 3' 2' Γ Every positive rational will occur in this list at least once. We throw out those that already appeared and thus get a correct listing This list indeed represents a bijection between N and Q : + 1 + 2 3 4 5 and so Q and N (or N) are numerically equivalent.! If X is dominated by Y but Y is not dominated by X (so that the Schröder-Bernstein theorem does not apply and X is not isomorphic to Y), we say that X is strictly dominated by Y and write X < Y.

The situation may be well visualized by the diagram describing the order relation of Example δ. There, we have no first element (because d and e cannot be compared) but we have two minimal elements, d and e. On the other hand, a is both a last element and a maximal element. Next we discuss the very important notion of bounds. Let X be a poset and A some subset of X. ) An element b G X is called a lower bound of the subset A iff for every α G A, we have b ^ a. Note that a subset A need not have any lower bound, and even if it has one, this b need not belong to A (it can be some element of X that is not contained in A).