By John E. Maxfield

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If you are afraid of heights above 15 feet? C la ssro o m E x e rc is e 2 - 3 . Set up a display of real dominos and use them to explain the idea of mathematical induction. Exercise 2-4 . Prove that A is a semigroup with binary operation + . 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.

Leaving out all detail, outline the successive enlargement of N to C. N stands for? , and so forth. 46 O T H E R A B S T R A C T A LG E B R A S Classroom Exercise 2-39. Give reasons for these “trivial” parts of defini tions: 1. A field must have at least two elements. 2. A rational number afb cannot have a zero denominator. 3. -J- is in <2i> but 3 is in /. 4. 999 • • • . What possible advantage can there be to such a complicated form for 3? 5. Why do we assume a ^ 0 in ax2 + bx + c = 0? Classroom Exercise 2-40.

It would be simpler just to add numerators and add denominators when adding fractions, without the bother of common denominators, but it would lead to situations like _ 12 2fi ' I 3G ----r,_5 which would not agree with our experience with the integers 3 and 2. If r is a solution in / to bx = a and s is a solution in / to dx = c, then r + s is a solution to (