Text six the limit

One of the principal concefrts of modern mathematics is the concept of the limit.

The sequence a₁, a₂, a₃ ... is said to have¹¹ the limit a as n tends to infinity if, corresponding to any positive number s, no matter how small, there may be found'¹ an integer (depending on e), such that

|a — a_n|<e

for all n >N.For example, the sequence whose n-th term is an= 1/n

has the limit 0 for increasing я:

1/n-> 0 as n->00

In fact, as we go out farther and farther in th

e sequence, the terms become smaller and smaller. If we go out far enough

in the seqience, we can be sure that each of its terms will differ from 0 by as little as we please. For any positive number

e we may choose any integer N greater than —.

A sequence a_n with a limit a is called convergent There is another way of expressing the limit concept.If lim a_n = a, and if we enclose a in the interior of an interval /, then no matter how small l may be, all the numbers a_n for which n is greater than or equal to some integer N will lie within /. At most N — l terms, namely, a_v a₂, ... a_N-1 can lie outside I. The statement lim a_n = a is equivalent to the statement: If I is any interval with a as its centre, then almost all of the numbers a_n lie within /.

If in a sequence a_n the members become so large that even­tually a_n is larger than any preassigned number K, then we say that a_n tends to infinity and write lim] a_n = oo, or a_n->oo. For example, n² ->oo and 2ⁿ ->oo if n ->oo. A sequence tending to infinity is still called divergent]

A sequence where a_n+1 > a_n for any n is said to be mono­tone increasing. For example, the sequence 1/2,1/3, ... n/(n+1) is monotone increasing. Similarly, a sequence for which a_n > > a_n+1 such as the sequence 1, 1/2, 1/3 ..., is called monotone

decreasing.

The behaviour of a monotone sequence is especially easy to determine.Any monotone increasing sequence that has an upper bound must converge to a limit. A similar statement holds for any monotone decreasing sequence with a lower bound

Obligatory Words and Word Combinations

limit (n), principal (a), infinity (n), no matter (how), In fact, term(n), differ (v), convergent (a), enclose (v), divergent (a), upper (lower) bound, converge (v), at most, monotone in­creasing (decreasing) sequence.

§ 2

The limit concept for functions is defined just as that of sequences of numbers. Namely, the function f (x) has the limit о as x tends to the value x₁ if, corresponding to every positive

number e, no matter how small, there may be found a positive number 8 (depending on e) such that

|f(x) —a|<e for all values of x satisfying the inequality

lx-x₁|<б The limit definition of functions makes it possible to introduse another important concept, the concept of the continuity of function with the help of the following defi-nition.The function f (x) is continuous for the value x = x₁ if f(x)->f (x₁) for x -> х₁,In other words the function f (x) is continuous for x = х₁ if; corresponding to every positive number e, no matter how small, there may be found a positive number 8 (depending on e) such that

If(x)-f(x₁)I<e

for all x satisfying the inequality

x—x₁|<б

In terms of the graph of a function и = / (x), the defini­tion of continuity takes the following geometrical form. Choose any positive number e and draw parallels to the x-axis at a height f (x1) — e and f (x₂ )+ e above it. Then it must be possible to find a positive number 8 such that the whole portion of the graph which lies within the vertical band of width 28 about x_l is also contained within the horizontal band of width 2e about f (x₁).

As an illustration of the general definition we shall analyse the function

We may restrict x to a fixed interval |x| < M where M is an arbitrarily selected number. Writing

F(x₁)-f(x)=

(l+x²)(l+x₁²) we find for|x|<M and |x,|<m

\H^xi)~f(x)\_<\x-x₁\ •|х+х₁|<|х-х₁|.Ш.

Hence it is clear that the difference on the left side will be smaller than any positive number e if only

(perhaps very small) interval J of length 28 with a as midpo- int such that the value of f (x) everywhere in J differs from f(a) by less than e Hence, since f (a) = 2e we can be sure that f (x) > e everywhere in J, so that F/ (x) > 0 in J. But the interval J is fixed, and if я is sufficiently large the little ■nterval I_n mast necessarily fall within J, since the sequence just as, namely (adv), continuity of a function, in terms. I_n tends to zero. This yields the contradiction; for it follows portion (n), vertical band, width (n), horizontal band, the way I_n was chosen that the function f (x) has opposite analyse (v), restrict (v), select (v). from the way I endpoints of every I_n, so that f (x:) must have

negative values somewhere in J. Thus the absurdity of f (a) > § 3 > 0 and (in the same way) of f (a) < 0 proves that f (a) = 0.

Obligatory Words and Word Combinations

of

A continuous function of a variable x which is positive for some value of x and negative for some other value of 1 in a closed interval a <=x<=b of continuity must have the value zero for some intermediate value of x.Thus, if f (x) is continuous as x varies from a to b,"while f (a) < 0 and f (b) > 0, then there will exist a value a of x such that a < a < b and

Proof. We consider the interval l : a < x < b in which the function f (x) is defined, and|bisect"it by marking the mid

point, х₁ = (a+b)/2 If at this mid-point we find that f (x) =0

then there remains nothing further to prove.If, however, f (x1) <> 0, then f (x₁) must be either greater than or less than zero. In either case one of the halves of I will again have the property that the sign of f(x) is different at its two ex< tremes.

Let us call this interval I ₁ We continue the process by bisecting .Then either f (x) = 0 at the mid-point, of l₁, or we can choose an interval I₂, half of I₁, with the property that the sign f (x) is different at its two extremes.

(Repeating this procedure, either we shall find after a defi­nite number of bisections a point for which f(x) — til of we shallobfain a sequence of nested intervals I _1, I₂, /₃.../;.... In the latter case, the Dedekind-Cantor postulate assures the exisftne of a point a in I common to all these intervals. We assert that f (a) = 0, so that a is the point whose existence proves the theorem.

We shall prove that f (a) = 0 by assuming the contrary and deducing a contradiction. Suppose that f (а) <> 0, e. g. thet f(a ) = 2e > 0. Since f (x) is continuous, we can find

Obligatory Words and Word Combinations

closed (a), midpoint (n), extreme (n), assert (v), bisection (n), assure (v), existence (n), deduce (v), opposite (a), absurdity (n), intermediate (a), in the same way.

TEXT SEVEN

THE DERIVATIVE

I*

1

The tangent is the limit of the secant, and the slope of he tangent is the limit of the slope of the secant. The slope of the secant t₁ given by the formula:

Suppose у = f (x) is a curve and P (x, y) is a point on this curve We consider on the curve another point P₁, near P, with coordinates x₁ y₁ The straight line joining P and P, we call t₄. It is a secant of the curve, which approximates to the tangent at P when P₁ is near P. The angle from the x-axis to t we call a₁. Now if we let x₁ approach x, then P₁ will move along the curve toward P, and the secant t₁ will ap­proach as a limiting position the tangent t to the curve at P. If a denotes the angle from the x-axis to t, then, as x₁ -»x The tangent is the limit of the secant, and the slope of the tangent is the limit of the slope of the secant.

The slope of the secant t_t is given by the formula:

slope of

2Ь

Now, after the cancellation, there is no longer any diffi­culty with the limit as x₁ -> x. The limit is obtained by «sub­stitution»; for the new form x₁ +x of the difference quotient is continuous and the limit of a continuous function as x_L -> x is simply the value of the function for x_t = x. In our case x + x = 2x, so that

In a similar way we can prove that

for f (x) = X^s we have f' (x) = 3x². In general, for

f(x) = xⁿ

a hori; where n is any positive integer, we obtain the derivative

The simplest non-trivial example is the differentiation el the function

which accounts to finding the slope of a parabola. This is the

the second derivative f" (x) i important both in analysis and geometry. Expressing the rate of change of the slope simplest case that teaches us how to carry out the passage f'(x) of the

If we denote the operation of forming a difference by the symbol A, we have

slope of

when

Slope of t = limit of. slope

the limits are evaluated as x_l -» x, i. e. as Ax = x_x — x ->0 The slope of the tangent t to the curve is the limit of the difference quotient Ay/Aх as Ax = x_l — x approaches 0 The original function f (x) gave the height of the curve у = f (x) for the value x. We may now consider the slope of the curve for a variable point P with the coordinates x and y which we denote by f' (x) and call the derivative of the function ! (x).

The limiting process by which it is obtained is called diffe, rentiation of f (x). This process is an operation which attaches!

to a given function f (x) another function f' (x)

Obligatory Words and Word Combinations

derivative (n), join (v), secant (n), approximate (v), tan gent (n), approach (v), limiting (a), slope (n), evaluate (v) differentiation (n), difference quotient.

§ 2

for all values of x This also follows from the definition.

Examples.The simplest function is f (x) = с where с is a constant. The graph of the function у =f (x) = с is a hori zontal line coinciding with all its tangents. It is obvious that

fo the limit when the result is not obvious from the outset. We have

If we should try ''•to pass to the limit directly in numerator and denominator we should obtain the meaningless expres­sion 0/0. But we can avoid this by rewriting the difference quo­tient and cancelling, before passing to the limit, the dis­turbing factor x₁ — x. In evaluating the limit of the difference quotient we consider only values x₁ <> х. Thus we obain the expression:

Obligatory Words and Word Combinations

coincide (v), parabola (n), passage (n), obvious (a), directly (adv), numerator (n), meaningless (a), expression (n),cancel (v), no longer, substitution (n), in a similar way, in general.

31

curve у = f (x), the second derivative gives an

§ 3

indication of the way the curve is.bent,If f" (x) is positive in an interval then the rate of change of f' (x) is positive. A positive rate of change of a function means that the values of the function increase as x increases. Therefore f" (x) > О means that the slope f'_: (x) increases as x increases, so that the curve becomes steeper where it has a positive 'slope and less steep where it has a negative slope. We say that the curve is concave upward.Similarly if f" (x) < 0, the curve y = = f (x) is concave downward.

The parabola у = f (x) = x² is concave upward everywhere because f"(x) = 2 is always positive. The curve у= f (x) = = x³ 1s concave upward for x >, 0 and concave downward for x < 0. This саn bе seen by its second derivative f" (x) = 6x. ... If s denotes the arc length along the curve, and a the slope angle, then a = h (s) will be a function of s. As we trave; along the curve s = h (s) will change. The rate of change h (s) is called the curvature of the curve at the point where the arc length is s. (The curvature k can be expressed in terms of the first and second derivatives of the function у — [ (x) defining the curve:

Obligatory Words and Word Combinations

analysis (n), rate of change, indication (n),bend (v), con­cave upward (downward), arc(n), steep (a), curvature (n).

question can be decided if we form the second derivative, /"(x).The sign of this derivative indicates the convex or concave shape of the graph.Therefore, f" (x) > 0 at the mini­ma points of the function, ana f" (x) < 0 at the maxima points. The vanishing of the second derivative usually indicates a point of inflection at which no extremum occurs. As an example we corfsider the polynomial

/(x) = 2x³ —9x² + 12x+ 1, and obtain

f'[x) = 6x² — 18x + 12, f"{x) = 12*— 18. The roots of the quadratic equation f (x) = 0 are x_t =1, х_г = 2. At these points the values of the second derivative are ., f"(x) =-6<0, /"(x₂) = 6>0.

Hence f(x) has a maximum, f (x_t) — 6, and a minimum, f (x₂) = 5.

Obligatory Words and Word Combinations

maximum (n) (pi. maxima), minimum (n) (pi. minima), describe (v), latter (a), ascending (descending) curve, (indi­cate (v), convex (a),j vanish (v), inflection (n), extremum (n).

TEXT EIGHT DIFFERENTIAL EQUATIONS

§4

If we describe the curve у = f (x) in the direction of increas-

. ing values of x, then a positive derivative, f' (x) > 0 at a point means ascending curve (increasing values of y), a negative

derivative,f' (x) < 0 means descending curve, while f' (x) = 0

means a horizontal direction of the curve for the value x.

At a maximum or minimum, the slope must be zero.

Thus we can find the maxima and minima of a [given func-tion by first forming f' (x), then finding,the values for which this derivative vanishes and finally investigating which of these values furnish maxima and which minima. The latter

§ 1

A differential

A differential equation for an unknown function и = f (x) with derivative u' = f' (x) is an equation involving u_{, 1,} and possibly the independent variable х, as for example

и' = и + sin (xu) or

и' + Зu = x².

Mоrе generally, a differential equation may involve the second derivative, u" = f" (x), or higher derivatives as in 'he example

u" + 2u'— 3u=0.

Un any case the problem, is to find a function u =f (x) that satisfies the given equation.

Solving a differential equation is a wide generalization of the problem of integration in the sense of finding the pri­mitive function of a given function g (x), which amounts to solving the simple differential equation

u' = g(x). For example, the solutionsof the differential equation u' = x²

are the functions и = x³ /3+ с where с is any constant.

Obligatory Words and Word Combinations

differential equation, involve (v), integration (n), primitive function, amount (to) (v).

§2

The diferential equation

has as a solution the exponential function и = e^x, since the exponential function is its own derivative More generally, the function и = се", where с is any constant, is a solution of equation u' =u. Similarly, the function

и = ce^kx, where с and k are any two constants is a solution of the diffe­rential equation

u' = ku.

■ b, where it is some constant.

But ln(u)/k is a primitive function of 1/(ku), so that x = h (u)

Conversely, any function и = f (x) satisfying equation u' = ku must ,be of the form ce^kx For if x = h (x) is the inverse function of и = f (x), then according to the rule for finding the derivative of an inverse function we have

Hence

and

Setting e-^bk (which is a constant) equal to c, we have a = ce^kx, as wos be proved

the great significance of the differential equation u' = ku lies in the fact that it governs physical processes in which a trantity и of some substance is a function of the time t,

and in which the quantity u ,is, changing at each instant at a rate proportional to the velue of и at that instant. In such a case the rate of change at the instant t

is equal to ku, where к is a constant, к being positive if и is increasing, and negative if и is decreasing. In either case, и satisfies the differential equation u' = ku; hence

The constant с is determined if we know the amount u₀ which was present at the time t — 0. We must obtain this amount if we set t = 0

so that

Note that we start with the knowledge of the rate of change of и and deduce the law (u = u₀e^kt) which gives the actual amount of u at any time у This is just the inverse of the prob­lem of finding the derivative of a function.

Obligatory Words and Word Combinations

exponential function, inverse function, instant (n), quanti­ty (n), amount (n), note (v).

$ ³

A typical example is that of radioactive disintegration Let и =j (t) be the amount of some radioactive substance at the time t; then on the hypothesis*' that each individual

35

particle of the substance has a certain probability of disinte-grating in a given time, and that the probability is unaffected by the presence of other such particles, the rate at which ц is disintegrating at a given time t will be proportional to u, -i. e. to the total amount present at that_r time. Hence и will satisfy the equation u' = ku with a negative constant k that measures the speed of the disintegration process, and therefore

и = u₀ е^kt. It follows that the fraction of и which disintegrates in two equal time intervals is the same; for if u_l is the amount present at time t₁ and u₂ the amount present at some later time t₂, then

which depends only on t₂ — t₁. To find out how long it will take for a given amount of the substance to disintegrate until only half of it is left, we must determine s = t_t — t₁ so that

from which we find

For any radioactive substance, the value of sis called the half-life period, and s or some similar value such as value t

It follows that

for which u₂/u₁=999/1000) can be found by experiment For radi­um, the half-life period is about 1550 years and

Obligatory Words and Word Combinations

disintegration (n),substance (n), particle (n), hypothesis (n)i probability (n), fraction (n) speed (n), find out (v), radium (n)¹ half-life period.

TEXT NINE

THE INTEGRAL

§1

The fist basic concept of the calculus is that of integral.Here we shall understand the integral as an expression of the area under a curve by means of a limit.If a positive conti­nuous function у = f (x) is given,.e. g. у = x² or u= 1 +cos x then we consider the domain bounded below by the segment on the x-axis from a coordinate a to a greater coordi­nate b, on the sides by the perpendiculars to the x-axis at these points, and above by the curve у — f (x).Our aim is to calculate the area A of this domaion

Since such a domain cannot,in general, be decomposed into rectangles or triangles, no immediate expression of this area is available for explicit calculation. But we can find an approximate value for A and. thus represent A as a limit in the following way: We subdivide the interval from x =a to x = b into a number of small subintervals erect perpen-diculars at each point of subdivision and replace each strip of the domain under the curve by a rectangle whose height is chosen somewhere between the greatest and the least height of. the curve in that strip.The sum S of the areas of these rectangles gives an approximate value for the actual area A under the curve. The accuracy of this approximation will be better the larger the number of rectangles and the smaller the width of each individual rectangle. Thus we can characte­rize the exact area as a limit: If we form a sequence,

S_v S_v S₃, ...

of rectangular approximations to the area under the curve in such a manner thet the width of the widest rectangle in S_n tends to 0 as n increases, then the sequence (1) approaches the limit A

(2)

and this limit A,the аrеа under the curve is independent of theparticular way in which the sequence (1) is chosen. so 1оng as the width of the approximating rectangles tend to zero, for ixample s_n can arise from S _n-1 by adding one or more new points of subdivision for S_n-1 or the choice of points of subdivision for S„ can be entirely independent of

37

the choice for Sn_i. The area A of the domain, expressed by this limiting process, we call by definition the integral of the function f (x) from a to b. With a special symbol, «the integral sign». it is written

(3)

choose as the height of each approximating rectangle the value of y = f (x) at the right-hand endpoint of the subinterval Then the sum of the areas of these rectangles will be

which is abbreviated as

The symbol J, the "dx" and the name «integral» were intro­duced by Leibniz in order to suggesf the way in which the limit is obtathfed To explain this notation we shall repeat in more detail the process of approximation to the area A. At the same time the analytic formulation of the limiting process will make it possible to discard the restrictive assump­tions f (x) > 0 and b > o, and finally to eliminate the prior intuitive concept of area as the basis of our definition of integral.

Obligatory Words and Word Combinations

calculus (n), integral (n), bound (n), by means of, calcu­late (v), in general, decompose (v), rectangle (n), be available, explicit (a), erect (v), accuracy (n), manner (n), suggest (v), to detail, restrictive (a), finally (adv), eliminate (v).

Let us subdivide the interval from a to b into n small subintervals, which, for simplicity only, we shall assume to be of equal width, (b-a)/n We denote the points of subdivision by

We introduce for the quantity

the difference between consecutive x-values, the notation Aх (read «delta x»),

where the symbol A means «difference». (It is an «operator symbol, and must not be mistaken for a number). We ma)

read «sigma from j = 1 to я») means the

Here the symbol

sum of all the expressions obtained by letting j assume in turn the values 1, 2, 3, ... n.

Now we form a sequence of such approximations S„ in which n increases indefinitely so that the number of terms in each sum (5) increases, while each single term f (xj) Ax approaches 0 because of the factor Ax (b — a)/n.As n increases, this sum tends to the area A

(6)

Obligatory Words and Word Combinations

subdivide (v), consecutive (a), mistake (v), in turn, because of, approximation (n), right-hand (left-hand), single (a).

TEXT TEN

THE FUNDAMENTAL THEOREM OF THE CALCULUS

. To formulate"⁷ the fundamental theorem we consider the integral of a function у = f (x) from the fixed lower limit a to (he variable upper limit x. To avoid confusion between the upper limit of integration x and the variable x that appears in the symbol f (x) we write this integral in the form

(1)

indicating that we wish to study the interval as a function F(x) of the upper limit xn This function F (x) is the area under the curve у = f (и) from the point и = a to the point и = x. Sometimes the integral F (x) with a variable upper limit is called as «indefinite» integral.

Now the fundamental theorem of the calculus is: The derivative of the indefinite interval (1) as a function of x is equal to the value of f (u) at the point x.

In ether words,the process of integration leading from the -function f (x) to 'F.{x).> ^is undone,[inverted, by the process of differentiation, applied to F (x)

Oh an intuitive basis the proof is very easy It depends on the interpretation of the integral F (x) as an area, and would be obscured³³ if one tried to represent F (x) by a graph and the derivative F' (x) by its slope" Instead of this original geometrical interpretation of the derivative we retain the geometrical explanation of the integral F (x)but proceed in an analytical way with the differentiation of F (x).\The diffe­rence

F(_Xl)-F(x)

is simply the area between x and x₁ and we see that this area lies between the values (x_L — x) m and (х_г — x) M,

(x_l — x)m< F(х — F(x)< (x₁ — x) M,

where M and in are respectively the greatest and the least values of f (и) in the interval between x and x₁ For these two products are the areas of rectangles including the curved area and included in it, respectively. Therefore

We shall assume that the function f (u) is continuous so that if x_l approaches x, then M and m both approach f (x). ШеПсе we have

We shall assume that the function f (u) is contmuous so that if x_x approaches x, then M and m both approach / (x). Hence we have

(2) as stated.

as stated.

Obligaiory words ana word combinattons

fundamental theorem of the calculus, avoid (v), invert (v), apply (v), instead of, explanation (n).

40

Obligatory Words and Word Combinations

fundamental theorem of the calculus, avoid (v), invert (v),

flnnlv (ч\ instead nf pvnlanatinn 1гл\

§2

The fundamental theorem of the calculus may be formula- ted in another way:

F (x), the integral of f (u) with fixed lower Iimit and a variable upper limit x, is a primitive function of f(x)

We say «a» primitive function and not «the» primitive function, for it is immedialy clear that if G (x) is a primitive function of f (x), then

f f (x) = G (x) + с (с — any constant) is also a primitive function, since H' (x) = G' (x). The con­verse, is also true. The primitive functions, G (x) and H (x), can differ only by a constant. For the difference U (x) = G (x) — — И (x) has the derivative U' (x) = G'(x)—H' (x) = f (x) — f(x) = 0, and is therefore constant, since a function repre­sented by an everywhere horizontal graph must be constant.

This leads to a most important rule for finding the value of an integral between a and b provided we know a primitive function G (x) of f (x), Ассоding to our main theorem,

is also a primitive function of f(x). Hence F(x)=G(x)+c where с is constant. The constant с is determtned it we remember that F (a) =Sf(u) du = 0. This gives 0 = G (a) + c

or if we write ft instercf or x,

so that с = — G (a). Then the definite integral between the limits a and x will be

irrespective of what particular primitive [unction 0 (x) we have chosen. In other words,

To evaluate the definite integral J f (x) dx, we need only

find a function G (x) such that G (x) = f (x) and then form the difference G (b) — g(a)

TEXT ELEVEN

GROWS AND RINGS

Consider a set S of elements, which we denote by a, ft, с

A law by which, given any ordered pair of elements a and b of S, possibly not distinct we can derive a unique element c, is called a law of composition for S. A non-acuous set S of elements with a law of composition which satisfies certain conditions, explained below, is called a group.

We denote the element resulting from the combination of a and b (in the given order) by aft; if the resulting element of S is с we write

ab =c.

ab is a uniquely defined element of S, but may be different from ba.

The law of composition is said to be associative¹¹ if, given any three elements a, ft, с of S, we have the equation

(aft) с = a (bc).

We then write this element as abc. The conditions that a set 5, with a given law of composition, should form a group are that

(I) the law of composition is associative;

(II) given any two elements a, b of S, there exist elements x, у such that

ax = b and ya = b.

As an example of a group, consider the possible derange­ments of the integers 1, 2, 3. If a, p, у is any derangement of these integers, we denote the operation of replacing 1, 2, 3 by a, p, у by the symbol

If, in a given group, the equation xy = yx is always true, we say the group is commutative, or Abelian. A very simple example of such a group is provided by the natural integers (positive, zero and negative), the law of composition being ordinary addition of integers"

It Is often convenient, when dealing" with commutative groups, to use the symbol of addition for the law of composi-

If, in a given group, the equation xy = yx is always true, we say the group is commutative, or Abelian. A very simple example of such a group is provided by the natural integers (positive, zero and negative), the law of composition being ordinary addition of integers

It is often convenient, when dealing" with commutative groups, to use the symbol of addition for the law of

42

-

tion, writing a + b instead of ab. We then cail the group an additive group. It is important to remember that this nota­tion is never used for a non-commutative group.

Obligatory Words and Word Combinations

group (n), ring (n), distinct (a), law of composition, non-vacuous (a), result (v), deal (with) (v), additive group, de­rive (v), Abelian commutative group, convenient (a).

§2

Let b be any element of S. Then there exist elements c, i such that

From condition (I) we have

We now obtain certain properties common to all groups. From condition (II) we know that, given any element a, there exist elements e and / such that

In particular, in the first of these equations put b = f, and in the second put b = e. Then f ~ fe and fe = e. Hence e = f. If there is another e' with the properties of e,

e'e = e', and e'e = e and therefore e is unique. We have thus established the exist­ence of a unique element e of the group such that ae = a = ea

for every element of the group. This element e is called the unity of the group. In the case of an additive group it is usual­ly called the zero of the group, and denoted by 0. Now we consider the equation

ax=e,

where a is any element of the group, e being the unity, By (II) 'his equation has a solution x. Then

xax =xe=x.

43

43

and therefore the element f = xa has the property fx = x, for the x considered. But, by an argument used above, fb = b for any b in the group. In fact, let с be an element satisfying the equation xc — b. Then

It follows, taking b = e, that / = e. Therefore xa = e. If у is any element such that

ay = e = ya, then

у = ye = yax = ex = x.

Hence x is uniquely defined by the equations

ax = e = xa.

This element is called the inverse of a, and is denoted by a~^l. (In the case of an additive group it is called the negative of a, and denoted by —a. We then write b — a for b + (—a)). We now show that the equations

ax = b, ya = b, where a, b are any elements of the group, serve to define x and у uniquely. For

x = ex = a^-1ax = a^-1b, and

у = ye = yaa~^l= ba~^x. Hence x and у are determined explicitly. In particular, the equation

a^_,x = e

has a unique solution, but

a^-1a = e. The solution is therefore x = a. Therefore

(a-¹)^-1 = a. In the case of an additive group this becomes

— (— a) = a.

A non-vacuous subset s of S may, with the law of composi­tion assumed for the elements of S, also form a group. This is called a subgroup of the given group. The following conditions are evidently necessary and sufficient for the elements of s to form ⁷⁸ a subgroup:

(I) if s contains elements a, b, it contains ab;

ill) if s contains an element a, it also contains ar^l,

Obligatory Words and Word Combinations

in particular, unity (n), above (adv), serve(v),evident (a).

§3

A set of elements may have more than one law of compo­sition. We shall be particularly concerned with sets having two laws of composition, under one of which the set forms a commutative group. We write this group as an additive group, and refer to the corresponding law as the addition law of the set. The zero of the group is called the zero of the set.

The second law of composition is called the multiplication law, and the result of combining elements a, b of the set by this law is denoted by the product ab. Multiplication need not be commutative, but we shall require it to be associative'³. It is said to be distributive over addition if

a ib + c) = ab+ ac, and (b + c) a = ba + ca,

for all a, b, с in the set.

A ring, then, is a set of elements with two laws of composi­tion, addition and multiplication, with the properties: (I) the set is an additive group with respect to⁸³ addition; (II) multiplication is associative and distributive over addition.

The following examples will illustrate the various possibili­ties which may arise in the study of rings. In the first four cases the laws of composition for the elements involved'³ are addition and multiplication as usually defined: I — The set of all complex numbers. II — The set of all integers, positive, zero, and negative.

3. — The set of all even integers.

4. — The set of all integers, reduced modulo the integer in.

V — The set of all matrices " of a and columns whose elements are complex numbers.

Obligatory Words and Word Combinations

combine (v), distributive (a), possibility (n), even integer, reduce (v), matrix (n) (pi. matrices), column (n).

44

45

§4

Let R and R* be two rings such that to each element of a of R there corresponds a unique element a* of R* and such that any element a* of R* arises from exactly one ele­ment a of R. Such a correspondence is said to be one-to-one. Now suppose, in addition , that the correspondence is such that if a, b correspond respectively to a*, b*, then a + b and aft correspond respectively to a* + b* and to a*b*. The correspondence is then called an isomorphism.

Isomorphism between rings is a relation of the class known аs equivalence relations ⁸². Consider any set S of elements a, p,

у and let there be a relation, which we denote by ,

between the elements of S, so that, given any two elements a, p, we know whether a ~ p is true or false, if the relation ~ is:

(I) reflexive, that is, a ~ a for all a in S;

(II) symmetric, that is a ~ p implies p ~-a;

(III) transitive, that is, if a ~ p and P ~ 7, then a ~ f; we say it is an equivalence relation.

An equivalence relation between the elements of a set S divides S into subsets⁸⁸, no two of which have any elements in common⁸³. If a, p are the same subset, a ~ p. Every element of S lies in one of these subsets.

It is clear that if S is in the set of all rings, and if a ~ p means that the ring a is isomorphic with the ring p, the rela­tion is an equivalence relation. We often speak of two isomor­phic rings as being equivalent, implying that if in our discus­sion we replace one ring by the other (making any necessary consequential substitutions), nothing in our conclusions is altered.

A subset S of the elements of a ring R is said to forma subring of R if the elements of S form a ring under the addition and multiplication laws of R. For this to be¹³ so it is necessary and sufficient that if a and b arc any two elements of S, then a — b and aft belong toS.

If S is a subring of R, R is said to be ² an extension of S. The following theorem is frequently used.

If A and B* are two rings, and A is isomorphic with a subring В of B", there exists an extension Л* of Л which is isomorphic with B*, this isomorphism including that between A and B".

Obligatory Words and Word Combinations

one-to-one, in addition, isomorphism (n), equivalence relation, reflexive (a), symmetric (a), transitive (a), in common, discussion (n), alter (v), belong (v), extension (n).

§5

Let R be any ring, l a subset of R with the following pro­perties:

(I) if a, b are in l, then a — b is in l;

(II) if a is in l, r in R, then ra is in l.

We call l a left-hand ideal in R; a right-hand ideal is defined by reading ar instead of ra in (II). The two ideals coincide when multiplication in R is commutative.

To show that if l is not vacuous it is a subring of R we remark that if r is in l conditions (I) and (II) coincide with the condmons stated above for a subring. On the other hand^8S, not all subrings are ideals, since the condtlon (II) need not

R itself is an ideal in R, and is called the unit ideal^8a. The subset of R which consists only of the zero of R is also an ideal. These two ideals are usually called the improper ideals of R. A ring containing a pfoper ideal is the ring of integers (Example II above). If m is any integer greater than 1, the set I of all integers of the form ± ma, where a is an integer, clearly forms a proper ideal in this ring.

Obligatory Words and Word Combinations

left-hand (right-hand) ideal, on the other hand, unit ideal, proper (improper) ideal.

TEXT TWELVE