《Discrete Mathematics II教学-华南理工》Review Group.pptVIP

Abstract Algebra Special Elements Definition: Let ? be a binary operation on a set A. An element e ∈ A is an identity element for ? if for all a ∈ A, a ? e = a = e ? a. Definition: Let ? be a binary operation on a set A. An element θ ∈ A is an zero element for ? if for all a ∈ A, a ? θ = θ = θ ? a. Definition: Let ? be a binary operation on A and suppose that e is its identity. Let x be an element of A. An inverse of x is an element y ∈ A such that x ? y = e = y ? x. Examples Let ? = + on Z. 0 is the identity element. No zero and inverse element. (2) Let ? = ? on Q . Then 1 is the identity element and 0 is the zero element. Every nonzero element x has a multiplicative inverse 1/x Semigroups（半群） Definition: A semigroup consists of a set on which is defined an associative binary operation. Definition: A semigroup (A, *) is said to be commutative (or Abelian) if the binary operation is commutative. Example The set of natural numbers, with the operation of addition, is a commutative semigroup. Monoids（独异点） Definition: A monoid consists of a set on which is defined an associative binary operation with an identity element. A semigroup is a monoid if and only if it has an identity element. Example The set N of natural numbers with the operation of multiplication is a commutative monoid. Indeed the operation of multiplication is both commutative and associative, and the identity element is the natural number 1. Groups A group (G,·) is a nonempty set G together with a binary operation · on G such that the following conditions hold: (i) Closure: For all a,b in G, the element a · b is a uniquely defined element of G. (ii) Associativity: For all a,b,c in G, we have a · (b · c) = (a · b) · c. (iii) Identity: There exists an identity element e in G such that e · a = a ? ? and ? ? a · e = a for all a in G. (iv) Inverses: For each a in G there exists an inverse element a-1 in G such that a · a-1 = e ? ? and ? ? a-1 · a = e. A group (G, · ) is said to