intro abstract algera - p.garret

[...]... was before attempting to prove it! This is typical of this sort of argument! In some circumstances, a seemingly di erent proof concept works better: Well-Ordering Principle Every non-empty subset of the positive integers has a least element This Well-Ordering Principle sounds completely innocuous, but it is provably logically equivalent to the Principle of Induction Another logically equivalent variant... of ordered n-tuples a1 ; a2 ; : : : ; an  of elements ai of A Some very important examples of cartesian powers are those of R or Q or C, which arise in other contexts as well: for example, R2 is the collection of ordered pairs of real numbers, which we use to describe points in the plane And R3 is the collection of ordered triples of real numbers, which we use to describe points in three-space The... 2; 2; 4g Of course, no one often really operates this way A function f : A ! B is surjective or onto if for every b 2 B there is a 2 A so that f a = b A function f : A ! B is injective or one-to-one if f a = f a0  implies a = a0 That is, f is injective if for every b 2 B there is at most one a 2 A so that f a = b A map is a bijection if it is both injective and surjective The number... that then A = B as sets Since the intersection is non-empty, there is some element y 2 A B Then, by the de nition of equivalence class", for all a 2 A we have a  y, and likewise for all b 2 B we have b  y By transitivity, a  b This is true for all a 2 A and b 2 B , so since A and B are equivalence classes we have A = B Done A set S of non-empty subsets of a set S whose union is the whole set... school': x4 + 4y4 has a factorization into two quadratic pieces, each with 3 terms! Find this factorization Hint: x4 + 4y4 = x4 + 4x2 y2 + 4y4 , 4x2 y2 = x2 + 2y2 2 , 2xy2 4 2 Induction and the Well-ordering Principle The meaning of the word `induction' within mathematics is very di erent from the colloquial sense! First, let P n be a statement involving the integer n, which may be true or false... integer exponent k? 8 3 Sets Sets and functions Equivalence relations 3.1 Sets Here we review some relatively elementary but very important terminology and concepts about sets and functions, in a slightly abstract setting We use the word map as a synonym for function", as is very often done Naively, a set is supposed to be a collection of `things' ? described by `listing' them or prescribing them by a... that this is not a terribly precise description, but will be adequate for most of our purposes We can also say that a set is an unordered list of di erent things There are standard symbols for some often-used sets: = fg = set with no elements Z = the integers Q = the rational numbers R = the real numbers C = the complex numbers A set described by a list is something like S = f1; 2; 3; 4; 5; 6; 7; 8g which... Sets can also be elements of other sets For example, fQ; Z; R; Cg is the set with 4 elements, each of which is a familiar set of numbers Or, one can check that ff1; 2g; f1; 3g; f2; 3gg is the set of two-element subsets of f1; 2; 3g The intersection of two sets A; B is the collection of all elements which lie in both sets, and is denoted A B Two sets are disjoint if their intersection is If the intersection... grammatically correct sentence, but are making no general claims about whether the sentence is true, true for one particular value of n, true for all values of n, or anything It's just a sentence Now we introduce some notation that is entirely compatible with our notion of function, even if the present usage is a little surprising If the sentence P n is true of a particular integer n, write P n =... absolutely no requirement that a function be described by any recognizable or simple `formula' 10 As a silly example of the formal de nition of function, let f : f1; 2g ! f2; 4g be the function `multiplyby-two', so that f 1 = 2 and f 2 = 4 Then the `o cial' de nition would say that really f is the subset of the product set f1; 2g  f2; 4g consisting of the ordered pairs 1; 2; 2; 4 That is, formally