[PPT模板]Chapter3-Binary System and Boolean Algebra.ppt

科技英语中数学公式的读法 数学公式的读法(Pronunciation of mathematical expressions) 1 逻辑(Logic) there exists for all p q p implies q / if p, then q pq p if and only if q p is equivalent to q p and q are equivalent 2 集合(Sets) xA x belongs to A / x is an element (or a member) of A xA x does not belong to A / x is not an element (or a member) of A AB A is contained in B / A is a subset of B ABA contains B / B is a subset of A ABA cap B / A meet B/ A intersection B ABA cup B/ A join B / A union B B/A A minus B/the difference between A and B ABA cross B / the Cartesian product of A and B(A与B的笛卡 尔积) 数学公式的读法(Pronunciation of mathematical expressions) 数学公式的读法(Pronunciation of mathematical expressions) 3 实数(Real numbers) x+1x plus one x-1x minus one x1x plus or minus one xyxy / x multiplied by y (x-y)(x+y)x minus y, x plus y x over y =the equals sign x=5x equals 5 / x is equal to 5 x5x (is) not equal to 5 xyx is equivalent to (or identical with) y 3 实数(Real numbers) xyx is greater than y xyx is greater than or equal to y x yx is less than y xyx is less than or equal to y 0x1zero is less than x is less than 1 0x1 zero is less than or equal to x is less than or equal to 1 |x|mod x / modulus x x2x squared / x (raised) to the power 2 x3x cubed x4 x to the fourth / x to the power four xn x to the nth / x to the power n x-n x to the (power) minus n 数学公式的读法(Pronunciation of mathematical expressions) 数学公式的读法(Pronunciation of mathematical expressions) 3 实数(Real numbers) n! n factorial (x+y)2 x plus y all squared xi xi / x subscript i / x suffix i / x sub i the sum from i equals one to n ai / the sum as i runs from 1 to n of the ai x over y all squared x hat x barx tilde 数学公式的读法(Pronunciation of mathematical expressions) 4 线性代数(Linear algebra) |A|the norm (or modulus) of x OA / vector OA OA / the length of the segment OA ATA transpose / the transpose of A A-1A inverse / the inverse of A 数学公式的读法(Pronunciation of mathematical expressions) 5 函数(Functions) f(x)fx / f of x / the function f of x f : STa function f from S to T x maps to y / x is sent (or mapped) to y f(x) f prime x / f dash x / the (rst) derivative of f with respect to x f(x)f doubleprime x / f doubledash x / the second derivative of f with respect to x f(x)f tripleprime x / f tripledash x / the third derivative of f with respect to x f(4)four x / the fourth derivative of f with respect to x ln ylog y to the base e / log to the base e of y / natural log (of) y 数学公式的读法(Pronunciation of mathematical expressions) 5 函数(Functions) the partial (derivative) of f with respect to x1 the second partial (derivative) of f with respect to x1 the integral from zero to infinity the limit as x approaches zero from above Computer English Chapter 3 Binary System and Boolean Algebra Key points:Key points: u useful terms and definitions seful terms and definitions of Binary system and Boolean of Binary system and Boolean AlgebraAlgebra Difficult points: Difficult points: C Conversion onversion of the Binaryof the Binary Systems and Systems and Boolean AlgebraBoolean Algebra Requirements:Requirements: 1. Concepts of Number System and their conversion 2. Boolean Algebra 3. Moores Law New Words it should be noted that he was also a philosopher. Leibnizs reasons for advocating the binary system seem to have been mystical. He felt there was great beauty in the analogy between zero, representing the void, and one, representing the Deity. 一位十七世纪的德国数学家Gottfrid Wilhelm von leibniz提倡二进制，只 使用0和1两个符号作为基数。这么一位杰出的数学家提倡用如此简单的 数字似乎有些奇怪。应该注意到他还是一位杰出的哲学家。Leibniz提倡 二进制的原因似乎有些神秘。他感到用零代表虚无和一代表上帝之间的 相似之处很有优势。 3.2 The Binary System Regardless of how good Leibnizs reasons for advocating it were, the binary system has become very popular in the last decade. Present-day digital computers are constructed to operate in binary or binary-coded number system, and present indications are that future machines will also be constructed to operate in these system. 不管leibniz提出的理由有多好，二进制在过去的十年里变得很流行。现 在的数字计算机以二进制原理构成，当前的迹象表明将来的机器将还构 造成按这些系统操作。 3.2 The Binary System The basic elements in early computers were relays and switches. The operation of a switch, or relay, can be seen to be essentially binary in nature. That is the switch is either on (1) or off (0). The principal circuit elements in more modern computers are transistors similar to those used in radios and television sets. The desire for reliability led designers to use these devices so that they were essentially in one of two states, fully conduction or non-conducting. A simple analogy may be made between this type of circuit and an electric light. 早期计算机的基本元件是继电器和开关，开关或继电器的操作可以看作 是基本的二进制性质。那就是开关是开(1)或关(0)两种状态。更先进一些 的计算机主要电器元件是类似于收音机和电视机的晶体管。可靠性的要 求使设计者采用这些装置，他们基本上处于两个状态之一，完全导通或 截止。在这种电路和电灯之间可以做简单的模似。 3.2 The Binary System At any given time the light (or transistor) is either on (conducting) or off (not conduction). Even after a bulb is old and weak, it is generally easy to tell if it is on or off. The same sort of thing may be seen in radios. As a radio ages, the volume generally decreases, and we compensate by turning the volume control up. Even when the radio becomes very weak, however, it is still possible to tell easily whether it is on or off. 任意给定的时间里，电灯（或晶体管）处于导通或截止状态之一。即使 电灯泡很旧了，一般很容易区分开它是开或关的状态。同样的事情也可 以在收音机中看到，当收音机老化了，音量就减低了，我们用调高音量 来补偿。即使当收音机很旧了，还能容易区分是开或关状态。 3.2 The Binary System Because of the large number of electronic parts used in computers, it is highly desirable to utilize them in such a manner that slight changes in their characteristics will not affect their performance. The best way of accomplishing this is using circuits which are basically bistable (having two possible states). 由于计算机中大量采用了电子器件，强烈要求利用他们的一些特性，即 当特性稍有变化时不至影响性能。做到这些的最好方法是采用双稳态电 路。 3.3 Boolean Algebra New Words the intersection of two sets consists of those elements that are in both given sets. We use the symbol to denote the union, and to denote the intersection of two sets. For example, if B=b, d, e, then BS=a, b, c, d, e, and BS=b. 于是，如上定义的集合S就有一个它的补集（相对于集合T）。任何两个集合（已 给定集合的若干子集）的并集包含了出现于这两个子集中某一个集合或同时出现 于这两个集合中的所有元素；两个集合的交集包含了同时出现于这两个集合中的 元素。我们用符号“”来表示两个集合的“并（运算）”，用“” 来表示两个集合 的“交（运算）”。例如，如果B=b,d,e,那么，BS=a,b,c,d,e,BS=b。 3.3 Boolean Algebra While other set operations may be defined, the operations of complementation union and intersection are of primary interest to us. A Boolean algebra is a finite or infinite set of elements together with three operationsnegation, addition, and multiplicationthat correspond to the set operations of complementation, union, and intersection, respectively. Among the elements of a Boolean algebra are two distinguished elements: 0, corresponding to the empty set; and 1, corresponding to the universal set. 虽然我们可以定义其他一些集合运算，但求补、并和交运算是我们最感兴趣的三 个集合运算。一个布尔代数就是一个有限集或无限集，以及建立在该有限集或无 限集上的三种运算否定、加或乘，这三个运算分别对应于集合的求补、并和 交运算。在布尔代数的元素中有两个特殊的元素：0，对应于空集；1，对应于全 集。 3.3 Boolean Algebra For any given element of a Boolean algebra, there is a unique complement a with the property that a+a=1 and aa=0. Boolean addition and multiplication are associative and commutative, as are ordinary addition and multiplication, but otherwise have somewhat different properties. The principal properties are given in Table 3-2, where a, b, and c are any elements of a Boolean algebra. 对于一个布尔代数中任意给定元素a，都有一个唯一的补a，它满足a+a=1和aa=0 。布尔加和布尔乘与普通的加和乘一样，满足结合律和交换律，但除此之外含有 一些不太相同的特性。其主要特性由表3-2给出，其中a, b和c是一个布尔代数中的 任意元素。 3.3 Boolean Algebra Table 3-2 Distributivity 分配律 a(b+c)=ab+ac a+(bc)=(a+b)(a+c) Idempotency 同一律 a+a=a aa=a Absorption laws 吸收律 a+ab=a a(a+b)=a DeMorgans laws德摩根定理 (a+b)=ab (ab)=a+b 3.3 Boolean Algebra Since a finite set of n elements has exactly 2n subsets, and it can be shown that the finite Boolean algebras are precisely the finite set algebras, each finite Boolean algebra consists of exactly 2n elements for some integer n. For example, the set algebra for the set T defined above corresponds to a Boolean algebra of 32 elements. 由于n个元素的有限集有且只有2n个子集，而且很显然有限布尔代数一定是有限集 合代数，所以对某个整数n而言，每个有限布尔代数也有且只有2n个元素。例如， 上文定义的集合T的集合代数就对应一个有32个元素的布尔代数。 3.3 Boolean Algebra While it is possible to use a different symbol to denote each element of a Boolean algebra, it is often more useful to represent the 2n elements of a finite Boolean algebra by binary vectors having n components. With such a representation the operations of the Boolean algebra are accomplished componentwise by considering each component as an independent two- element Boolean algebra. This corresponds to representing subsets of a finite set by binary vectors. 虽然我们可以用不同的符号来表示布尔代数中的每一个元素，但最常用的方法是 用一个有n个分量的二进制向量来表示一个有限布尔代数的2n个元素。用这样一种 表示方法，布尔代数的所有运算都以分量的形式完成，而每一个分量被认为是一 个独立的二值布尔代数。这种做法对应于用二进制向量来表