Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 6 Abstract algebra Groups Field Lattics and Boolean algebra Rings 6.1 Operations on the set Definition 1:An unary operation on a nonempty set S is a function f from S into S; A binary operation on a nonempty set S is a function f from S×S into S; A n-ary operation on a nonempty set S is a function f from Sn into S. closed Associative law: Let * be a binary operation on a set S. a(bc)=(ab)c for a,b,cS Commutative law: Let * be a binary operation on a set S. a*b=b*a for a,bS Identity element: Let * be a binary operation on a set S. An element e of S is an identity element if a * e = e * a = a for all a S. Theorem 6.1: If * has an identity element, then it is unique. Inverse element: Let * be a binary operation on a set S with identity element e. Let a S. Then b is an inverse of a if a * b = b * a = e. Theorem 6.2: Let * be a binary operation on a set A with identity element e. If the operation is Associative, then inverse element of a is unique when a has its inverse Distributive laws: Let and be two binary operations on nonempty S.For a,b,cS, a(bc)=(ab)(ac), (bc)a=(ba)(ca) Associative law commutative law Identity Inverse elements element + √ √ 0 -a for a √ √ 1 1/a for a0 Definition 2: An algebraic system is a nonempty set S in which at least one or more operations Q1,…,Qk(k1), are defined. We denoted by [S;Q1,…,Qk]. [Z;+] [Z;+,*] [N;-] is not an algebraic system Definition 3: Let [S;*] and [T;] are two algebraic system with a binary operation. A function from S to T is called a homomorphism from [S;*] to [T;] if (a*b)=(a)(b) for a,bS. Theorem 6.3 Let be a homomorphism from [S;*] to [T;]. If is onto, then the following results hold. (1)If * is Associative on S, then is also Associative on T. (2)If * is commutative on S, then is also commutation on T (3)If there exist identity element e in [S;*],then (e) is identity element of [T;] (4) Let e be identity element of [S;*]. If there is the inverse element a-1 of aS, then (a-1) is the inverse element (a). Definition 4: Let be a homomorphism from [S;*] to [T;]. is called an isomorphism if is also one-to-one correspondence. We say that two algebraic systems [S;*] and [T;] are isomorphism, if there exists an isomorphic function. We denoted by [S;*][T;](ST) 6.2 Semigroups,monoids and groups 6.2.1 Semigroups, monoids Definition 5: A semigroup [S;*] is a nonempty set together with a binary operation * satisfying associative law. Definition 6: A monoid is a semigroup [S;*] that has an identity. Let P be the set of all nonnegative real numbers. Define & on P by a&b=(a+b)/(1+a*b) Prove[P;&]is a monoid. 6.2.2 Groups Definition 7: A group [S;*] is a monoid, and there exists inverse element for aS. (1)for a,b,cS,a*(b*c)=(a*b)*c; (2)eS,for aS,a*e=e*a=a; (3)for aS, a-1S, a*a-1=a-1*a=e [R-{0},] is a group [R,] is a monoid, but is not a group [R-{0},], for a,bR-{0},ab=ba,, Abelian (or commutative) group Definition 8: We say that a group [G;]is Abelian (or commutative) group if ab=ba for a,bG. [R-{0},],[Z;+],[R;+],[C;+] are Abelian (or commutative) group . Example: Let [G;*] be a group with identity e. If x*x=e for xG, then [G;*] is an Abelian group. Example: Let G={1,-1,i,-i}. 1 -1 i -i 1 -1 i -i 1 -1 i -i -1 1 -i i i -i -1 1 -i i 1 -1 Multiplication table Abelian group G={1,-1,i,-i}, finite group [R-{0},],[Z;+],[R;+],[C;+],infinite group |G|=n is called an order of the group G Let G ={ (x; y)| x,yR with x 0} , and consider the binary operation ● introduced by (x, y) ● (z,w) = (xz, xw + y) for (x, y), (z, w) G. Prove that (G; ●) is a group. Is (G;●) an Abelian group? [R-{0},] , [R;+] a+b+c+d+e+f+…=(a+b)+c+d+(e+f)+…, abcdef…=(ab)cd(ef)…, Theorem 6.4: If a1,…,an(n3), are arbitrary elements of a semigroup, then all products of the elements a1,…,an that can be formed by inserting meaningful parentheses arbitrarily are equal. a1*a2*…*an n ai i 1 If ai=aj=a(i,j=1,…,n), then a1*a2*…*an=an。 na Theorem 6.5: Let [G;] be a group and let aiG(i=1…,n). Then (a1…an)-1=an-1…a1-1 Theorem 6.6: Let [G;] be a group and let a and b be elements of G. Then (1)ac=bc, implies that a=b(right cancellation property)。 (2)ca=cb, implies that a=b。(left cancellation property) S={a1,…,an}, al*aial*aj(ij), Thus there can be no repeats in any row or column Exercise p333 9,10,11,18,19,22,23, 24; P340 5—7,13,14,19—22 P357 1,2,6-9, Prove Theorem 6.3 (2)(4)