Basic Concepts in Algebra x1. Notations and terminologies (1.1) Some symbols • 8: \for all" • 9: \there exists" • 7!: maps to (under a map/function) • lhs := rhs (\rhs" is the definition of \lhs") • X r S: the complement in S of a subset S ⊆ X • 2S, where S is a set: the set of all subsets of S • Z: the set of all integers • N: the set of all natural numbers (including 0) • Q: the set of all rational numbers • R: the set of all real numbers • C: the set of all complex numbers • N>0: the set of all positive integers • M2(R): the set of all 2 × 2 matrices with entries in R • GL3(Q): the set of all invertible 3 × 3 matrices A with entries in Q, i.e. there exists a 3 × 3 matrix B with entries in Q such that A · B = B · A = I3, where I3 is the 3 × 3 matrix whose diagonal entries are 1 and off-diagonal entries are 0. • GL2(Z): the set of all invertible 2 × 2 matrices A with entries in Z such that there exists a 2 × 2 matrix B with entries in Q satisfying A · B = B · A = I3 ; this amounts to the condition that the determinant of A is ±1. • R[x]: the set of all polynomials with coefficients in R • Q[x; y]: the set of all polynomials in two variables x and y with all coefficients in Q • Z[x; y; z]: the set of all polynomials in three variables x; y; z with all coefficients in Z. (1.2) Definition Let f : S ! T be a map from a set S to a set T . 1 (a) The map f is said to be injective (or, f is an injection) if for any two elements s1; s2 2 S, f(s1) = f(s2) only if s1 = s2. Another standard terminology for the same concept: f is one-to-one. (b) The map f is said to be surjective (or, f is a surjection) if for every element t 2 T , there exists an element s 2 S such that f(s) = t. In other words the image f(S) of f is equal to the target T of the map f. Another standard terminology for the same concept: f is onto. (c) The map f is said to be bijective (or, f is a bijection) if f is both injective and surjective. Another standard terminology for the same concept: f is a one-to-one and onto correspondence. (1.3) Definition Let S1;:::;Sn be sets. The product S1 × · · · Sn of S1;:::;Sn is the set consisting of all n-tuples (x1; : : : ; xn) such that xi 2 Si for all i = 1; : : : ; n. Such a product Qn Q is also denoted i=1 Si, where \ " is the symbol for a product. The above definition in words can also be expressed by the following formula n Y Si = S1 × · · · Sn := f(x1; : : : ; xn) j xi 2 Si 8 i = 1; : : : ; ng i=1 (1.4) Remark We will not need to use the product of an infinite family of sets. In case you wonder what such an infinite product is, suppose that I is an indexing set, and Si is a family of sets indexed by I. (a) The disjoint union ti2I Si of the sets Si. is the set of all pairs (i; x) where i is an element of the indexing set I and x is an element of the set Si indexed by i. For each j 2 I, the set Sj is naturally identified with the subset of all elements in ti2I Si of the form (j; x) such that x 2 Sj. (Another notation form the disjoint union is \`" because \coproduct" of sets are ` nothing but disjoint union, so ti2I Si can also be written as i2I Si. However we want to avoid such notation because the coproduct of a finite family of groups is the same as their products, but the coproduct of an infinite family of groups is a proper subgroup of the product group.) Q (b) The product i2I Si is the set of all functions f : I ! ti2I Si from I to the disjoint union ti2I Si of the sets Si such that (1.5) Definition Let f : X ! S and g : Y ! S be maps of sets. The fiber product of f : X ! S and g : Y ! S, denoted by X ×f;S;g Y or X ×S Y for short, is the subset of X ×Y 2 consisting of all pairs (x; y) 2 X × Y such that f(x) = g(y). Denote by π1 : X ×S Y ! X and π2 : X ×S Y ! Y the two \projections", defined by π1(x; y) = x ; π2(x; y) = y 8(x; y) 2 X ×S Y: Clear f ◦ π1 = g ◦ π2 by construction. The triple (X ×S Y; π1; π2) satisfies the following universal property: For any maps of sets u: T ! X, v : T ! Y such that f ◦ u = g ◦ v, there exists a unique map h: T ! X ×S Y such that u = π1 ◦ h and v = π2 ◦ h. (1.6) Definition An equivalence relation on a set S is a subset R ⊆ S × S satisfying the following properties. • (R is reflexive) (x; x) 2 R for all x 2 S. In word, every element of S is equivalent to itself under the equivalence relation R. • (R is symmetric) If (x; y) 2 R then (y; x) 2 R. In words, if x is equivalent to y then y is equivalent to x. • (R is transitive) If x; y; z are elements in S such that (x; y) 2 R and (y; z) 2 R, then (x; z) in R. In words, if x is equivalent to y and y is equivalent to z, then x is equivalent to z. (1.7) Definition Let R be an equivalence relation on a set S. (a) The equivalence class under R containing an element x 2 R is the set of all elements y 2 S such that (x; y) 2 R. (So an equivalence class is a subset of S. Note that any two equivalence classes are either identical or disjoint. So the equivalence relation R partitions the set S into a disjoint union of equivalence classes.) (b) The set S=R is the set of all equivalence classes in S with respect to R. (So each element of the set S=R is a subset of S, i.e. S=R is a set of subsets of S.) Example: Two integers a; b are said to be congruent modulo 37 (notation: a ≡ b (mod 37)) if their difference is divisible by 37. Being congruent modulo 37 is an equivalence relation on Z. The set of all equivalence classes for this equivalence relation denoted by Z=37Z. Note that each element of Z=37Z is a subset of Z; one such element is 1 + 37Z, consisting of all integers n such that get 1 as the remainder if you divide n by 37. Of course you can replace 37 by any integer N and define an equivalence relation in a similar way. (1.8) Definition A partial ordering on a set S is a relation on S, written a b if this relation holds for the ordered pair (a; b), and the following properties hold. 3 (i) a a for all a 2 S. (ii) If a b and b c, a; b; c 2 S, then a c. (iii) If a; b 2 S, a b and b a, then a = b. A partial ordering on S is said to be a total ordering if property (iv) below holds. (iv) For any two elements a; b 2 S, either a b or b a. (1.9) Definition Let be a partial ordering on a set S. (a) An upper bound of a subset T is an element b 2 S such that t b for all t 2 T . (b) A maximal element of a subset T is an element m 2 T such that there is no element in T which is bigger than m. In other words, if t 2 T and m t, then t = m. (1.10) Zorn's Lemma. Let S be a non-empty partially ordered set such that every totally ordered subset T ⊂ S has an upper bound in S. Then there exists a maximal element in S. Zorn's Lemma is an equivalent form of the axiom of choice in set theory, which is known to be independent of the basic axioms in standard set theory and is consistent if the standard set theory is, i.e. assuming it will not lead to contradiction unless the standard set theory already does (an inconceivable scenario|then almost all of the known mathematics will have to be abandoned). Most mathematicians use Zorn's lemma freely. (1.11) Definition (a) Two sets S1 and S2 are said to have the same cardinality if there exists a bijection between S1 and S2; we write Card(S1) = Card(S2) if this is the case. (b) We say that the cardinality of a set S1 is less than or equal to a set S2 if there exists an injection from S1 ! S2. This property is equivalent (under the axiom of choice) to the existence of a surjection from S2 to S1. Notation: Card(S1) ≤ Card(S2). (1.12) Basic facts about cardinality, assuming the axiom of choice. (i) If Card(S1) ≤ Card(S2); and Card(S2) ≤ Card(S3), then Card(S1) ≤ Card(S3).