By Keith J. Devlin

The objective of this booklet is to supply the scholar starting undergraduate arithmetic with a great starting place within the simple logical techniques valuable for many of the themes encountered in a school arithmetic direction. the most contrast among most faculty arithmetic and collage arithmetic lies within the measure of rigour demanded at collage point. quite often, the recent scholar has no adventure of totally rigorous definitions and proofs, with the end result that, even if useful to address really tricky difficulties in, say, the differential calculus, he/she is completely misplaced while provided with a rigorous definition oflimits and derivatives. In influence, which means within the first few weeks at collage the scholar must grasp what's almost a complete new language {'the language of mathematics'} and to undertake a completely new mode ofthinking. remember the fact that, purely the very ablest scholars come via this strategy with out a good deal of difficulty.

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0} is a set which has ONE member. ) A set A is called a subset of a set B if every element of A is a member of B. For example, {t, 2} is a subset of {l, 2, 3}. We write 36 SETS, FUNCTIONS AND LOGIC A~B to mean that A is a subset of B. 2 (1) List all subsets of the set {I, 2,3, 4}. (2) List all subsets of the set {I, 2, 3, {I, 2} }. (3) Prove (by induction) that a set with exactly n elements has 2ft subsets. 2 Operations on sets There are various natural operations we can perform on sets. ) Given two sets A, B we can form the set of all objects which are members of either one of A and B.

6) Let A be any non-empty set. Define I A : A -+ A by We call IA the identity function on A. It is clearly bijective. 6 (1) Let A = {I, 2}, B = {I, 2, 3}. List all the functions from A to B. (2) List those functions in question (1) which are (a) one-one (b) onto (c) bijective. (3) Define f : flA -+ flA by f(x) = t(x + Ixl). Evaluate f(O), f(1), f(2). Is f one-one? Is f onto? Justify your answers. (4) Define f : flA -+ flA by f(x) = {X2 + 1,. if x ~ 0 x-I, If x < 0 f is one-one. Is f onto? f : JV -+ JV by Prove that (5) Define f( n) = {2n, if n is even n, I' fn 'IS 0 dd Show that f is one-one.

Define g : B ..... A by the rule: for each beB, g(b) is the unique element a of A for which f(a) = b. Clearly, g is as required for invertibility of f Since this is the only possible definition of g, we see also that g is unique. QED. The unique function g as above is called the inverse of f, and denoted by f - 1. 9 (1) Let f : A -+ B (2) Define be invertible. Show that f : ~ -+ ~ by f(x) = {X2 + 1, x + 1, if x ~ 0 if x < 0 Show that f is a bijection and find f - 1. (3) Define f: ~2 ..... [JI2 by f(x, y) = (x + 2y, x - y).