Group A group,denoted by (G,),is a set G with a binary operation GXG>G such that -Associativity:a(boc)=(ab)oc(associative) -Existence of identity:there exists e Gs.t.VxeG,ex=x oe =x(identity) -Existence of inverse:for any x e G,there exists y Gs.t.x y=yox=e (inverse) ·A group(G,)is commutative if Vx,.y∈G,xoy=yo X. Examples:(Z,+),(2,+),(Q{0},×),(R,+),(R{0}, X) 6
6 Group A group, denoted by (G, ◦), is a set G with a binary operation ◦: G×GG such that ─ Associativity: a ◦ (b ◦ c) = (a ◦ b) ◦ c (associative) ─ Existence of identity: there exists e ∈ G s.t. ∀x ∈ G, e ◦ x = x ◦ e = x (identity) ─ Existence of inverse: for any x ∈ G, there exists y ∈ G s.t. x ◦ y = y ◦ x = e (inverse) A group (G, ◦) is commutative if ∀x, y ∈ G, x ◦ y = y ◦ x. Examples: (Z, +), (Q, +), (Q\{0}, ×), (R, +), (R\{0}, ×)
Integers modulo n (1/2) Letn≥2 be an integer Definition: a is congruent to b modulo n,denoted as a b mod n, if n(a-b),i.e.,a and b have the same remainder when divided by n Definition: [al {all integers congruent to a modulo n) [al,is called a residue class modulo n,and a is a representative of that class. 7
7 Integers modulo n (1/2) Let n ≥ 2 be an integer Definition: a is congruent to b modulo n, denoted as a ≡ b mod n, if n|(a-b), i.e., a and b have the same remainder when divided by n Definition: [a]n = {all integers congruent to a modulo n} [a]n is called a residue class modulo n, and a is a representative of that class