By John E. Maxfield

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C. MacDuffee, Introduction to Abstract Algebra, Wiley, 1940, pp. 1-5; or E. ) Countable Infinity The natural numbers taken as a whole have the peculiar property that they can be put into one-to-one correspondence with a proper subset of themselves. ) A classic illustration of the peculiarities of arithmetic in such a case is that of the hotel with rooms numbered 1, 2, 3, . . , one room for each natural number. Suppose every room is full, but another guest arrives. The manager simply gives him room No.

A mapping from a set S to a set T is a relation on S to T such that each member of S is related to exactly one member of T. A mapping is onto if each member of T is related to at least one member of S. Since a binary operation is a mapping (Definition 1-1), it can be defined in terms of relations, too, as a relation on the set S x S to the set S. For instance, the operation addition in N, thought of as a relation, is the set of elements [(m, n), m + n], where m and n are in N. It contains ((2, 3), 5) and ((17, 1), 18) but does not contain ((1, 1), 1) or ((m, n)9 1) for any m and n in N.

Use mathematical induction on c to prove the associative law, a + {b + c) = (a + b) + c. You will need several applications of the inductive definition of addition. Which group properties fail to hold in the semigroup A? C la ssroom E x e rc is e 2 -5 . Have a classmate hold up his fingers separated into three bunches. Show how to demonstrate for a class how associativity of addition follows from counting. Demonstrate commutativity of addition. Prove that A is a semigroup with multiplication as binary Assuming the distributive law, prove by induction on k that 1<(mn).