数学专业英语[1]－The Real Number System

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数学专业英语－Mathematical Discovery

To give the flavor of Polya’s thinking and writing in a very beautiful but subtle case , a case that involve a change in the conceptual mode , I shall quote at length from his Mathematical Discovery (vol.II , pp.54 ff):

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EXAMPLE I take the liberty a little experiment with the reader , I shall state a simple but not too commonplace theorem of geometry , and then I shall try to reconstruct the sequence of idoas that led to its proof . I shall proceed slowly , very slowly , revealing one clue after the other , and revealing each gradually . I think that before I have finished the whole story , the reader will seize the main idea (unless there is some special hampering circumstance ) . But this main idea is rather unexpected , and so the reader may experience the pleasure of a little discovery .

A.If three circles having the same radius pass through a point , the circle through their other three points of intersection also has the same radius .

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Fig.1 Three circles through one point.

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This is the theorem that we have to prove . The statement is short and clear , but does not show the details distinctly enough . If we draw a figure (Fig .1) and introduce suitable notation , we arrive at the following more explicit restatement :

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B . Three circles k , l , m have the same radius r and pass through the same point O . Moreover , l and m intersect in the point A , m and k in B , k and l in C . Then the circle e through A , B , C has also the radius

Fig .2 too crowded .

Fig .1 exhibits the four circles k , l , m , and e and their four points of intersection A, B , C , and O . The figure apt to be unsatisfactory , however , for it is not simple , and it is still incomplete ; something seems to be missing ; we failed to take into account something essential , it seems .

We are dialing with circles . What is a circle ? A circle is determined by center and radius ; all its points have the same distance , measured by the length of the radius , from the center . We failed to introduce the common radius r , and so we failed to take into account an essential part of the hypothesis . Let us , therefore , introduce the centers , K of k , L of l , and M of m . Where should we exhibit the radius r ? there seems to be no reason to treat any one of the three given circles k ; l , and m or any one of the three points of intersection A , B , and C better than the others . We are prompted to connect all three centers with all the points of intersection of the respective circle ; K with B , C , and O , and so forth .

The resulting figure (Fig . 2) is disconcertingly crowded . There are so many lines , straight and circular , that we have much trouble old－fashioned magazines . The drawing is ambiguous on purpose ; it presents a certain figure if you look t it in the usual way , but if you turn it to a certain position and look at it in a certain peculiar way , suddenly another figure flashes on you , suggesting some more or less witty comment on the first . Can you recognize in our puzzling figure , overladen with straight and circles , a second figure that makes sense ?

We may hit in a flash on the right figure hidden in our overladen drawing , or we may recognize it gradually . We may be led to it by the effort to solve the proposed problem , or by some secondary , unessential circumstance . For instance , when we are about to redraw our unsatisfactory figure , we may observe that the whole figure is determined by its rectilinear part (Fig . 3) .

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This observation seems to be significant . It certainly simplifies the geometric picture , and it possibly improves the logical situation . It leads us to restate our theorem in the following form .

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C . If the nine segments

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KO , KC , KB ,

LC , LO , LA ,

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MB , MA , MO ,

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are all equal to r , there exists a point E such that the three segments

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EA , EB , EC ,

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are also equal to r .

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Fig . 3 It reminds you －of what ?

This statement directs our attention to Fig . 3 . This figure is attractive ; it reminds us of something familiar . (Of what ?)

Of course , certain quadrilaterals in Fig .3 . such as OLAM have , by hypothesis , four equal sided , they are rhombi , A rhombus I a familiar object ; having recognized it , we can “see “ the figure better . (Of what does the whole figure remind us ?)

Oppositc sides of a rhombus are parallel . Insisting on this remark , we realize that the 9 segments of Fig . 3 . are of three kinds ; segments of the same kind , such as AL , MO , and BK , are parallel to each other . (Of what does the figure remind us now ?)

We should not forget the conclusion that we are required to attain . Let us assume that the conclusion is true . Introducing into the figure the center E or the circle e , and its three radii ending in A , B , and C , we obtain (supposedly ) still more rhombi , still more parallel segments ; see Fig . 4 . (Of what does the whole figure remind us now ?)

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Of course , Fig . 4 . is the projection of the 12 edges of a parallele piped having the particularity that the projection of all edges are of equal length .

Fig . 4 of course !

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Fig . 3 . is the projection of a “nontransparent “ parallelepiped ; we see only 3 faces , 7 vertices , and 9 edges ; 3 faces , 1 vertex , and 3 edges are invisible in this figure . Fig . 3 is just a part of Fig . 4 . but this part defines the whole figure . If the parallelepiped and the direction of projection are so chosen that the projections of the 9 edges represented in Fig . 3 are all equal to r (as they should be , by hypothesis ) , the projections of the 3 remaining edges must be equal to r . These 3 lines of length r are issued from the projection of the 8^th , the invisible vertex , and this projection E is the center of a circle passing through the points A , B , and C , the radius of which is r .

Our theorem is proved , and proved by a surprising , artistic conception of a plane figure as the projection of a solid . (The proof uses notions of solid geometry . I hope that this is not a treat wrong , but if so it is easily redressed . Now that we can characterize the situation of the center E so simply , it is easy to examine the lengths EA , EB , and EC independently of any solid geometry . Yet we shall not insist on this point here .)

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This is very beautiful , but one wonders . Is this the “ light that breaks forth like the morning . “ the flash in which desire is fulfilled ? Or is it merely the wisdom of the Monday morning quarterback ? Do these ideas work out in the classroom ? Followups of attempts to reduce Polya’s program to practical pedagogics are difficult to interpret . There is more to teaching , apparently , than a good idea from a master .

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——From Mathematical Experience

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Vocabulary

subtle 巧妙的，精细的

clue 线索，端倪

hamper 束缚，妨碍

disconcert 使混乱，使狼狈

ambiguous 含糊的，双关的

witty 多智的，有启发的

rhombi 菱形（复数）

rhombus 菱形

parallelepiped 平行六面体

projection 射影

solid geometry 立体几何

pedagogics 教育学，教授法

commonplace 老生常谈；平凡的

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数学专业英语－Notations and Abbreviations (I) Learn to understand

* a- C' N7 K, K& o' n9 M

N set of natural numbers ) c. ~4 }1 T3 J3 l, w4 @2 O8 C2 q * X9 r4 L8 j: y! A1 J$ ? ; v3 B# q/ S. Y& K' G

3 F; T, R/ G1 d5 E9 A

Z set of integers 5 V- x! c* H5 f8 N4 N+ P1 {- z

, L# k/ Y- ?& _+ |+ r: e1 b+ ?

2 A& L* P' a2 ?# o& M

R set of real numbers / Z/ F9 `5 q+ \% `7 c) s

C set of complex numbers - D' C* m0 C! {

+ plus; positive ( H4 E/ `) [5 E6 n3 y# V

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－　　　　 minus; negative

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×　　　　 multiplied by; times

÷ divided by

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＝ equals; is equal to

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≡ identically equal to

≈,≌ approximately equal to % f/ ^3 I) Z; q& Y" R# I

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＞ greater than E9 K5 I' g% r( c, p

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≥ greater than or equal to - Z0 i' i! S/ m

＜ less than $ y6 }% _. J9 b5 e

% o0 K& u% N0 O# a

≤ less than or equal to

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》 much greater than

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《 much less than

square root ! q2 h' d" M% `6 I- [" X

5 S% l* {) E6 n R x

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cube root ; R, u' S4 K+ e% r8 D" ?

: O/ q, L7 R" Y. Q2 ^

nth root

4 F' K* O( ^+ c. o

' g* f+ j7 F' a$ D2 e: @2 e! V# q

│a│ absolute value of a 5 b4 B. ] Z9 ]( ? o8 C- a+ w

n! n factorial $ z9 M4 u5 D; f# w2 B

a to the power n ; the nth power of a ( y" {' i2 `6 d" }0 [4 M

2 d5 |9 X2 B8 X+ ~

[a] the greatest integer≤a

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the reciprocal of a % A- ^/ O& K( R2 ]

/ \3 I" I& g4 L1 \6 x

2 r+ _6 W$ d% B" g5 S3 D1 g6 g

2 k: i I" ?) \; y

Let A, B be sets

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∈　　　　 belongs to ; be a member of - W, W0 R" K! ^4 T

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not belongs to

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x∈A x os amember of A 7 {) C" l0 W9 l/ `5 q. m" `; [0 I

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∪ union

A∪B A union B

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∩ intersection

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A∩B A intersection B # J2 i' C! ?* w$ l! D! l

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A B A is a subset of B;A is contained in B & {, t* C7 d' Z1 W9 s

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A B A contains B

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complement of A 9 u" t9 G9 ~7 ~( g) n

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the closure of A

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empty set

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( ) i=1,2,…,r j=1,2,…,s r-by-s（r×s）matrix

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│ │I,j=1,2,…,n 　　　　determinant of order n 7 t" C; S3 g* Q# U8 w

$ t7 n ? a# w# ]% l0 u4 y

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det( ) the determinant of the matrix ( ) 8 P* s' p; @6 H) [, ~- Z4 C) L

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vector F 2 i4 G( f) f3 O! {/ R% M. [7 r' ]: \

x=( , ,…, ) x is an n-tuple of % P3 s E8 N9 a% u) }+ K5 u$ T

‖‖　　　　　　　　　　the norm of …

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‖ parallel to 4 Z) {: w; u4 p. O2 K( h% P; `

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┴ perpendicular to

the exponential function of x

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lin x the logarithmic function of x 1 o" T1 n& y' a; q

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2 p0 I3 `1 c }* a$ x

sie sine

& J6 c, j3 C. F/ [7 ~

cos cosine 0 }, R+ o: }2 a. b& [

tan tangent

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sinh hyperbolic sine

& g( F# A& q. O0 G3 z9 c0 d) b

cosh hyperbolic cosine

the inverse of f , q0 v& p5 O) E, H

f is the composite or the composition of u and v

the limit of …as n approaches ∞(as x approaches ) 0 y4 i7 E: o) A7 s3 l! R3 U

0 n. U! f; ~! U8 f7 y

x a x approaches a

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, the differential coefficient of y; the 1st derivative of y

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; Y k+ {; b) S; }4 q G( X3 ]

, the nth derivative of y : [8 _) X5 u& z( k( m$ S

$ y( U+ r! L4 a; m8 ?

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the partial derivative of f with respect to x

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3 N) l9 L8 z2 o/ Y, W( Y' ?4 S. u

the partial derivative of f with respect to y

+ ?/ Q N& ?6 Z6 N

the indefinite integral of f

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the definite integral of f between a and b (from a to b) , {; `) A5 n6 R# l: p) w

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the increment of x D/ D8 ]9 [, Q$ q# [% f1 [

differential x

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summation of …the sum of the terms indicated

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∏　　　　　　　　　　the product of the terms indicated

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=> implies 0 U4 N) O' H* v! X

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is equivalent to

（ ） round brackets; parantheses

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[ ] square brackets

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{ } braces

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1 C6 E$ ]1 ?0 m

- j. _' Z: s; y0 C3 S" Z5 Q) a1 K m1 Z

/ y' f E. G. c5 C O( g, u$ v8 U- l

hehe123

数学专业英语－Notations and Abbreviations (II) Learn to read # L+ S5 p: w. N1 F 2 W) w+ c: ], \' U% A

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0.1 0 point one; zero point one; nough point one 3 J* Z* L9 L1 i& M. a: V& w! `. Y + p7 g' \( q; \+ \4 j( b9 z% a 2 p+ t9 I* p( E' E

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0.01 0 point 0 one; zero point zero one

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4.9….. four point nine recurring 2 o! ?8 e0 t# K" R& G

8 @5 N/ a# C/ N, J( q! R4 ^3 C+ ~

3.03262626… three point nought three two six, two six recurring * S5 c' v2 F( e( L

8 ?0 M1 a. }+ l" Z/ D+ _

38.72 thirty eight point seven two

# t& t9 o ^ h3 l/ x: Y

j' W0 a, v: O+ `1 ~2 ^* w

a+b=c a plus b is c _: N* n! v, {" _! k9 ?

c-a=b c minus b is a ; b taken from c leaves a , F. T! K9 u. K! j: O

' {; ^8 C4 g- r6 M/ C: r! |0 R$ f8 Y

12÷3=4 twelve divided by 3 equals four

$ M5 O4 v: k2 k3 r

30=6×5 thirty is six times five

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6×5=30 six multiplied by five is thirty & F6 ~" s" e* A N

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7+3＜12 seven plus three is less than twelve , g" J: y2 {& `! {

4 a4 Y, G. \; d3 k6 O H% }3 d6 U

(a+b) bracket a plus b bracket closed 5 ]+ g+ K, L9 `+ x

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20:5=16:4 the ratio of twenty to five equals the ratio of sixteen to four ( F! C: S3 f4 J6 L$ [" `4 G

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a:b∷c:d a is to b as c is to d

0 T' `+ S9 V. Z/ c* ^: q

v=s/t v equals s divided by t; v is s over t 5 D# `$ X+ e. K+ K, ?# L) O

(a+b-c×d)+e=f a plus b minus c multiplied by d, all divided by e equals f

2 M4 e. ]/ T8 J$ f

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% percent

3/8% three-eighths percent

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x square; the square of x 3 \5 _/ _- m: _# i! E

" u; j p" d' R: }( k* i4 b! U( v

y cube; the cube of y

- l' g( H& M% `; v

the cube root of a

! q2 \2 e: w9 o; d

the square root of five hundred and eighteen

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the nth root of the difference of b and c % A+ D- `& I f2 A0 E

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lin x the natural logarithm of x; the Naperian logarithm of x , {9 Z/ d. o: A5 }! N' Y

log x to the base N

! e5 {! w5 B% a

4 }% K3 R$ p0 [; A( t+ k2 V

sigma % F3 P/ S- P! a

" a: E" {* H0 x4 x

b prime ! |; x5 N: l9 y: S0 n

* f/ L$ m4 Z& Z6 x* S+ o

b double prime

b triple prime

0 A0 t' {6 Y# s+ T" X

b sub m ; i" R; a2 |9 D) P3 Z

$ X& c+ L. |8 u' \/ j+ ~

b double prime sub n

# O1 r2 L Z1 c; v2 y

……… dots

- ?1 s. R& v3 c6 P) o, |# i8 B

π pi

5 @ m$ e7 Y$ i- u3 T

α alpha 1 n( a* e4 z. n }7 Q

. `4 I2 w3 f8 }0 Z

β beta / g5 |# p ~9 E2 D

γ,Γ gamma

5 c1 [) N5 F0 m2 `

+ Q: c2 G; q5 z1 y8 @5 }

δΔ delta $ V, P5 l" S4 {

ωΩ omega ; U O1 o1 U. F+ `4 `! j9 ^$ V4 K

ξ xi

1 W( g6 |0 M2 X( E5 r3 }

η eta

5 M2 n9 c W. w* |# h

; e" g2 k, O6 D7 | u4 W

ζ zeta ; J- n% o6 Y, u( p

# Q n9 ^. X$ b4 U- R# U

Φ phi - X/ d* o$ |+ M1 {, r

9 r1 o" p9 Z, i; i6 P

ψΨ psi

, Y7 U) j" R% y" I

χ chi z: t# t2 ~8 U& {! i: c! Y) c6 i

- `8 U3 P* b& I6 X4 R3 q

ρ rho

τ tau - {: T) }4 N2 O! u: T- `

, i9 X6 y/ [1 t. V% a% R

/ `# L, H1 P( W+ x" B3 E; _) \2 F

υ nu

μ mu 4 n: D5 i! B9 v) L$ {2 N0 N

' P! `3 e; K* a+ z2 E' A5 x+ n

λ lambda

" ?5 e* n1 v) E3 L) ^: J1 @8 P

3 P" }' V- E- k7 `* o

κ kappa ; a2 j& W, ^9 x; K8 K& v, A" z

+ ], b. l- J, Z5 z8 S

ε epsilon * U) J( Z1 F% V$ p( E# y

?1 ]) g* H g

θ theta

! O8 A) C3 ?# L R) o1 _. v& n* Q

∴ therefore

1 e/ ^( l1 `( x$ H

∵ because

there exist (s) M) s% H5 p, D4 K. T" E! ]( ]* ]

, ~" t: l+ T6 E4 R

for any

1 S$ Q7 q, J$ x+ [* W$ A8 U

$ J9 D9 b6 R$ [& I- o l6 u4 E

iff if and only if ' W: h$ j7 L- {& x

: Y9 p, h& {. M: o) Y. n' u' e

etc eccetra

e.g. for example

% K2 T6 S/ V) e( O

i.e. that is 8 a7 `4 a T1 P& N

, E& Q* J& q' y) [/ l, y) |- c

/ {$ Y- _" S- ?" T& \0 o/ S* C

viz. namely

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6 T2 c1 I; H2 }( o' V

w.r.t. with respect to . B2 ?' s2 k# Z, g# F$ m

4 ?+ P9 ~ ^; }6 k @8 Z( K

9 ?' r4 x9 G' Z& _8 N: P

# h) @. e, j* m. @3 K5 X

hehe123

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Vocabulary ( p* I7 F" z! Y6 ]( m* M2 d

7 N9 K" ^' {9 I4 U

/ L. Q. ^/ \$ K1 v1 d

real number 实数

* [& [+ Y w0 i" i3 L) c B M

negative 负的

) f4 L! e& T& u

the real number system 实数系

rational number 有理数

5 R1 a7 r& G' j/ @. P! Q

collection 集体,总体

ratio 比,比率

object 对象,目的

8 D: D% H- c3 _6 q: T& m$ `. Q

denominator 分母

) h8 k' S, R5 X- Z9 _. w* c% f. K$ H* ?& i

principle 原理,规则

numerator 分子

, V( v: p/ o" `9 h7 b3 D7 M: d& ~2 ?

adopt 采用

irrational number 无理数

. K( u8 z" g( Y) ]( `

define 定义(动词)

- A+ q% c1 Q8 `$ x

signify 表示

definition 定义(名词)

" u+ Y4 L/ W0 Q9 o. o

geometrical 几何的

establish 建立

straight line 直线

explicit 清晰的,明显的

+ E4 ]( L7 L0 x- n8 T$ f0 x. \

initial point 初始点

( {4 }9 J8 R) a: }4 ~) D! `

illustrate 说明

point of reference 参考点

0 r& T F& ^# k1 `

positive 正的

origin 原点

4 c* x: n/ a7 `' W

express 表达

assign 指定

plus 加

unit 单位

sign 记号,符号,正负号

property 性质

operation 运算,操作

7 ~) _5 g9 w( d. I$ M& c

closure property 封闭性质

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addition 加法

commutative 交换的

multiplication 乘法

6 }1 B3 |+ v2 L# M0 g8 B* m' |

associative 结合的

& O) p5 A a2 q! n

substraction 减法

parentheses 圆括号

6 c; L: `; s+ ]4 z8 G2 f. z

division 除法

brackets 括号

sum 和,总数

& B& s: x2 H, p) b

algebra 代数

procuct 乘积

0 R6 s [1 W6 A: L' E$ d8 Y

yield 产生

difference 差,差分

}: |" C/ b2 j1 q5 w

term 术语,项

$ ?+ V2 W8 ]- e4 e' v5 `* R, b

quotient 商

distributive 分配的

. ? Z" Y; N# @

symbolism 符号系统

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unique 唯一的

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minus 减

additive inverse 加法逆运算

identify 使同一

multiplicative inverse 乘法逆运算

count 计数

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reciprocal 倒数,互逆

natural number 自然数

concept 概念

zero 零

fraction 分数

integer 整数

2 G7 m3 x b" q) |4 E

arithmetic 算术的

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greater than 大于

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solution 解,解法

less than 小于

even 偶的

be equal to 等于

odd 奇的

3 P- S" v& b/ i. `

arbitrary 任意的

square 平方

absolute value 绝对值

square root 平方根

cube 立方

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induction 归纳法

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Note

1. Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system.

意思是:我们对数的实际工作经验使我们大家对支配着实数系的各原则早已有了某些熟悉,这里working作”实际工作的”解,govern作”支配”解.

2.The plus sign,”+”, used here not express the operation of addition, but is rather part of the symbolism for the numbers themselves.

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意思是:这里的正符号”+”不是表示加法运算,而是数本身的符号系统的一部分.

3. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero.

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这是定义数学术语的一种形式.下面是另一种定义数学术语的形式.

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A matrix is called a square matrix if the number of its rows equals the number of its columns.

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这里is called与is said to be 可以互用,注意is called后面一般不加to be而is said后面一般要加.

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4. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.

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与注3比较,这里用定语从句界定术语.

5. There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.

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意思是:存在唯一的一个实数,叫做零并记为0,具有性质a+0=0+a,这里(其中)a是任一实数.

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1) 这里called和denoted都是过去分词,与后面的词组成分词短语,修饰number.

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2) with the property是前置短语,修饰number.

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3) 注意本句和注3.中where的用法,一般当需要附加说明句子中某一对象时可用此结构.

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Exercise

I. Turn the following arithmetic expressions into English:

i) 3+(-2)=1 ii) 2+3(-4)=-10

iii) = -5 iv) =3

v)2/5-1/6=7/30

II. Fill in each blank the missing mathematical term to mark the following sentences complete.

i) The of two real numbers of unlike signs is negative.

ii) An integer n is called if n=2m for some integer m.

iii) An solution to the equation xⁿ=c is called the n is of c.

iv) If x is a real number, then the of x is a nonnegative real number denoted by |x| and defined as follows

x, if x 0

|x|= -x, if x <0

III. Translate the following exercises into Chinese:

i) If x is an arbitrary real number, prove that there is exactly one integer n such that x<n<x+1.

ii) Prove that there is no rational number whose square in 2.

iii) Given positive real numbers a1,a2,a3,…such that an<ca_n-1 for all n>2, where c is a fixed positive number, use induction to prove that an<c^n-1a₁, for all n>1.

iv) Determine all positive integers n for which 2ⁿ<n!

Ⅳ Translate the following passage into Chinese:

There are many ways to introduce the real number system. One popular method is to begin with the positive integers 1,2,3,…and use them as building blocks to construct a more comprehensive system having the properties desired. Briefly, the idea of this method is to take the positive integers as undefined concepts, state some axioms concerning them, and them use the positive integers to build a larger system consisting of the positive rational numbers. The positive irrational numbers, in turn, may then be used as basis for constructing the positive irrational numbers. The final step is the introduction of the negative numbers and zero. The most difficult part of the whole process is the transition from the rational numbers to the irrational num

Ⅴ. Translate the following theorems into English:

1. 定理A: 每一非负数有唯一一个非负平方根.

2. 定理B: 若x>0, y是任意一实数,则存在一正整数n使得nx > y.

Ⅵ. 1. Try to show the structure of the set of real numbers graphically.

2. List and state the laws that operations of addition and multiplication of real numbers obey.

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数学专业英语－Basic Concepts of the Theory of Sets 6 H5 P0 y1 \- b! `2 m Y* a 6 w) O3 E" I; g. I

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In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory. This subject, which was developed by Boole and Cantor in the latter part of the 19^th century, has had a profound influence on the development of mathematics in the 20^th century. It has unified many seemingly disconnected ideas and has helped to reduce many mathematical concepts to their logical foundations in an elegant and systematic way. A thorough treatment of theory of sets would require a lengthy discussion which we regard as outside the scope of this book. Fortunately, the basic noticns are few in number, and it is possible to develop a working knowledge of the methods and ideas of set theory through an informal discussion . Actually, we shall discuss not so much a new theory as an agreement about the precise terminology that we wish to apply to more or less familiar ideas.

In mathematics, the word “set” is used to represent a collection of objects viewed as a single entity

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The collections called to mind by such nouns as “flock”, “tribe”, ‘crowd”, “team’, are all examples of sets, The individual objects in the collection are called elements or members of the set, and they are said to belong to or to be contained in the set. The set in turn ,is said to contain or be composed of its elements.

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We shall be interested primarily in sets of mathematical objects: sets of numbers, sets of curves, sets of geometric figures, and so on. In many applications it is convenient to deal with sets in which nothing special is assumed about the nature of the individual objects in the collection. These are called abstract sets. Abstract set theory has been developed to deal with such collections of arbitrary objects, and from this generality the theory derives its power.

NOTATIONS. Sets usually are denoted by capital letters: A,B,C,….X,Y,Z ; elements are designated by lower-case letters: a, b, c,….x, y, z. We use the special notation

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X∈S

To mean that “x is an element of S “or” x belongs to S”. If x does not belong to S, we write x∈S. When convenient ,we shall designate sets by displaying the elements in braces; for example ，the set of positive even integers less than 10 is denoted by the symbol{2,4,6,8}whereas the set of all positive even integers is displayed as {2,4,6,…},the dots taking the place of “and so on”.

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The first basic concept that relates one set to another is equality of sets:

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DEFINITION OF SET EQUALITY Two sets A and B are said to be equal(or identical)if they consist of exactly the same elements, in which case we write A=B. If one of the sets contains an element not in the other ,we say the sets are unequal and we write A≠B.

SUBSETS. From a given set S we may form new sets, called subsets of S. For example, the set consisting of those positive integers less than 10 which are divisible by 4(the set{4,8})is a subset of the set of all even integers less than 10.In general, we have the following definition.

DEFINITION OF A SUBSET.A set A is said to be a subset of a set B, and we write

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A B

Whenever every element of A also belongs to B. We also say that A is contained in B or B contains A. The relation is referred to as set inclusion.

The statement A B does not rule out the possibility that B A. In fact, we may have both A B and B A, but this happens only if A and B have the same elements. In other words, A=B if and only if A B and B A .

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This theorem is an immediate consequence of the foregoing definitions of equality and inclusion. If A B but A≠B, then we say that A is a proper subset of B: we indicate this by writing A B.

In all our applications of set theory, we have a fixed set S given in advance, and we are concerned only with subsets of this given set. The underlying set S may vary from one application to another; it will be referred to as the universal set of each particular discourse.

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The notation

{X∣X∈S. and X satisfies P}

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will designate the set of all elements X in S which satisfy the property P. When the universal set to which we are referring id understood, we omit the reference to S and we simply write{X∣X satisfies P}.This is read “the set of all x such that x satisfies p.” Sets designated in this way are said to be described by a defining property For example, the set of all positive real numbers could be designated as {X∣X>0};the universal set S in this case is understood to be the set of all real numbers. Of course, the letter x is a dummy and may be replaced by any other convenient symbol. Thus we may write

{x∣x>0}={y∣y>0}={t∣t>0}

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and so on .

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It is possible for a set to contain no elements whatever. This set is called the empty set or the void set, and will be denoted by the symbolφ.We will consider φto be a subset of every set. Some people find it helpful to think of a set as analogous to a container(such as a bag or a box)containing certain objects, its elements. The empty set is then analogous to an empty container.

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To avoid logical difficulties, we must distinguish between the element x and the set {x} whose only element is x ,(A box with a hat in it is conceptually distinct from the hat itself.)In particular, the empty setφis not the same as the set {φ}.In fact, the empty set φcontains no elements whereas the set {φ} has one element φ(A box which contains an empty box is not empty).Sets consisting of exactly one element are sometimes called one-element sets.

UNIONS,INTERSECTIONS, COMPLEMENTS. From two given sets A and B, we can form a new set called the union of A and B. This new set is denoted by the symbol

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A∪B(read: “A union B”)

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And is defined as the set of those elements which are in A, in B, or in both. That is to say, A∪B is the set of all elements which belong to at least one of the sets A,B.

Similarly, the intersection of A and B, denoted by

A∩B(read: “A intersection B”)

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Is defined as the set of those elements common to both A and B. Two sets A and B are said to be disjoint if A∩B=φ.

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If A and B are sets, the difference A-B (also called the complement of B relative to A)is defined to be the set of all elements of A which are not in B. Thus, by definition,

A- B={X|X∈A and X B}

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The operations of union and intersection have many formal similarities with (as well as differences from) ordinary addition and multiplications of union and intersection, it follows that A∪B=B∪A and A∩B=B∩A. That is to say, union and intersection are commutative operations. The definitions are also phrased in such a way that the operations are associative:

(A∪B)∪C=A∪(B∪C)and(A∩B)∩C=A=∩(B∩C).

The operations of union and intersection can be extended to finite or infinite collections of sets.

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Vocabulary

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Set 集合 proper subset 真子集

Set theory 集合论 universal set 泛集

Branch 分支 empty set空集

Analysis 分析 void set 空集

Geometry 几何学 union 并，并集

Notation 记号，记法 intersection交，交集

Terminology 术语，名词表 complement余，余集

Logic 逻辑 relative to相对于

Logical 逻辑的 finite有限的

Systematic 系统的 disjoint不相交

Informal 非正式的 infinite无限的

Formal正式的 cardinal number基数，纯数

Entity 实在物 ordinal number序数

Element 元素 generality一般性，通性

Abstract set 抽象集 subset子集

Designate 指定， divisible可除的

Notion 概念 set inclusion 集的包含

Braces 大括号 immediate consequence直接结果

Identical 恒同的，恒等的

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Notes ( m+ M! w3 B/ w9 `1 F

1. In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory.

意思是：在讨论数学的任何分支时，无论是分析，代数或分析，利用集合论的记号和术语是有帮助的。

这一句中be it analysis, algebra, or geometry 是以be开头的状语从句，用倒装形式。类似的句子还有：

people will use the tools in further investigations, be it in mathematic, hysics , or what have you .

2. Actually, we shall discuss not so much a new theory as an agreement about the precise terminology that we wish to more or less familiar ideas.

意思是：事实上，我恩将讨论的与其说是一种新理论，不如说是关于精确术语的一种约定，我们希望将它们应用到或多或少熟悉的思想上去。

注意：not so much A as B 在这里解释为“与其说A不如说B。”类似的用法如：

This is not so much a lecture as a friendly chat.

(与其说这是演讲不如说是朋友间的交谈。)

3．Two sets A and B are said to be equal if they consist of exactly the same elements, in which case we write A=B.

数学上常常在给定了定义后，就 用符号来表达。上面句子是常见句型。类似的表达法有：

A set A is said to be a subset of a set B, and we write A=B whenever every element of A also belongs to B.

This set is called the empty set or the void set, and will be denoted by the symbol Φ.