复旦大学：《离散数学 Discrete Mathematics（上）》英文课件（赵一鸣）22/30

Chapter 6 Abstract algebra Groups v Rings√ Field√ a Lattics and boolean algebra

Chapter 6 Abstract algebra  Groups   Rings   Field   Lattics and Boolean algebra

6.1 Operations on the set o Definition 1: An unary operation on a nonempty set S is an everywhere function ffrom S into S; A binary operation on a nonempty set S is an everywhere function f from sXSinto S; A n-ary operation on a nonempty set s is an everywhere function f from Sn into S closed

6.1 Operations on the set ⚫ Definition 1：An unary operation on a nonempty set S is an everywhere function f from S into S; A binary operation on a nonempty set S is an everywhere function f from S×S into S; A n-ary operation on a nonempty set S is an everywhere function f from S n into S.  closed

Associative law: Let s be a binary operation on a set S. a*(b*c=(a*b)*c for Va,b,c∈S Commutative law: Let *k be a binary operation on a set s. a*b=b*a for Va, bES Identity element: Let be a binary operation on a set s. An element e of s is an identity element if=e*a=afor all a ∈S Theorem 6.1: If s has an identity element, then it is unique

 Associative law: Let  be a binary operation on a set S. a(bc)=(ab)c for a,b,cS  Commutative law: Let  be a binary operation on a set S. ab=ba for a,bS  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 s be a binary operation on a set S with identity element e Let a ES. Then b is an inverse ofa a*b b*ka=e o Theorem 6.2: Let a be a binary operation on a set a with identity element e. ifthe operation is associative, then inverse element of a is unique when a has its inverse

• 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 va,b,c∈S, a●(b*C)=(a·b*(a·),b*c)●a=(b●a)*(c●a) Associative law commutative Identity Inverse law elements element 0 -a for a l/a for ≠=0

 Distributive laws: Let  and • be two binary operations on nonempty S. For a,b,cS,  a•(bc)=(a•b)(a•c), (bc)•a=(b•a)(c•a) Associative law commutative law Identity elements Inverse element + √ √ 0 -a for a  √ √ 1 1/a for a0