1969-01-16 — Page 14

(Solutions for last week are omitted. If there is any

problem

please refer to Basic Concepts of New hawatles by Y. Lee, Essential Tutorial Press.)

南二个張四第

日八廿月一十年申戊歷要

WAH KIU YAT PC

報日橋

四期星日六十月一年九六九一曆公年八十五國民蓋中

育教帶華

") # 3x-2=0,- §1:37=2,

英文中學會考

|x log 3 == log 2

一九六九車中文物学会考試預習

0.3010

-0.631

4771

現代數學科

(+3)

#

數學科(十二) #144

b)

log (35-x3)

Log (5-x)

MODERN MATHEMATICS (11)

第十次預習題解答!

14)

(1)化簡下列各式:

(a) 3*** - 6x 3

3*(336)

3X3***

27-4=2 (32)

移項、化簡

31-2(5x379) — 36

34(3-3):

6(答)

(a-at) (b-b

ㄓ

23.

(c) Xligx_ 1000%

x=±0.631

4 log (35-X2) = 3 log (5-x) = log (5-x).

35-X3 5-x)=125-75x

15x-75X+90

13(x-2)(x-3)=

此两值均不能使35-285-X為負數或零 故能满足

答:

(A) logx loax log 1000 x

2 (logx) = log x-3=0

•1⁄2 (log 1000+ logx).

2+ logx

(log x + 1) (2 log x-3)=0.

log Li

107=10(10.

()t dog x+1=0 g logx

() Eicomitional

Any statement In the form. p. If and only if

q in called a biconditional, denoted by pa-q. The double arrow "" suggests the meaning of a combination of conditionals, na me 17 (p,q) A

1) A (a --> p) - Hence a biconditional can be also

a considered as a conjunction of two conditionals. The trith value of a biconditional is defined by its truth table as follows,

ра

A.ཐ

d?

TTT

F FF FT

ITF TAFF

F TFT

(hvimi “y, a biconditimal is true only.

when both of the components p and q have the

ske truth value, or both pq and

have the same true value,

Examples:

·(1) "2x + 4 - 16 if and only if x

biconditional is true because "if 2x+

then 2x+

than x- 6a and "if

are both me,

(2) "2 • 5 - 6 if and only if trees are.

insects

The

16."

The bisonditional is true, because both p

and are false.

"6. is odd if and only 15 is a prins number,

The biccnditional is false since the

components of the statements, ha ve different true values.

"Tautologies and contradictions

let F (p,q,r,s,...t) be the simbol

representing a composite statement which involves

a set of variable substatements p,q,ris, and so on

A proposition P(p,q,r,s,...t) is a tautology if

is true for any statements

Por Cor Tornet. Such propositions contain only T.

in the last columns of their truth tables.⠀⠀

Wi-the other hand, we may have a proposition which has only F in the last colum of its truth table. Such proposition is called a contradiction, In brevity, a proposition P(p,q,r,...t) is a contradiction if poi¶。iro to) is false for any statements P. You To and so on,

Examples (1) Prove each of the following

tautology.

fruth table of the roposition. JUBA (AUB)

FTTT

FEET T

TTF

The table tes only Thin the last colu

1s proved

Truth table of the pro

(AB),

ition, b)

FTTT

THEY

TT

T

hence

(b) 5×34 9×3-

(c) a+b2-a2

(1) 19 (a-a') (b~b')=ab+ab-af-ub

arb-az f2

af afz

(ab+ab1)(ab=ab

ab+a^l_(a+b+ab')

・原式

att

a2r ft

(abraf

(af+a'f')(a)

化蕙

a+b a b

-az-az f2)

◆解

(

· (ab+a'b') (ab-c08-)

log48+ log5-120g 1000

7dy log $48+ log 5 - log $100,0

log 12-1

2(big48+2 log 5 — log1000)

Log12-logio

103 10570

3x + L 3 x

·3*(3-1).

-3=0, Q long x

log 12-log 10

答:

1⁄2

log (405)

10.00

log 13

log 1.2 log 1.2

(d) 3x-5x

一素(答)

(AZ) 4524

分解

log 3

50

50

洛对数

x log 06

along 50

- 7.66.

(2)对數换底公式,得dog23

(logs 3+ logu?) (log 2+ logg4)

-=(22-2014) (Liga + log4)

log 3 2 log3)

=(

2143 x Abg2=4(*) (答)

Log

f) k log, 23+ logiz ! 效數字

(解)依对數换底公式,

查表

1.3617 1.1139 40414

12304

log 56=

logo b

1.6990

X=-0.766

77782

x, 求x之值至三位有

(e) {

log (xy) + log (7x-8y)-

23 13 Logr

lig!7 logx.

0.8451

1.3076+0.9053 — logx

2.8457

Rog 7

log x = 0.8451 X 22/29=== 1,8701

x=antilog 1.8701== 74.2 (*)

(2)解下列方程式,

(a) 2x27*-5x9x+3**_3*—0

(14) @ 27*—(33)x-(3*)* 代入,

1+1

分解

9x-(3%),

2× (3%) --5x (3x)*+3×3~ 3* -

2× (3*) — 5× (3*) + 2×3*—0

3* [2(3x)*~5(3)+2]=

3* (2×3* - 1) (3×2) =

*=0,則無意義,

log (x2+ y2) - log (x=xy+dg2)= |

(*) * (1) *, log (x-4) (7K 84) = log 100

(x-4) (7x~84)====100

7x=15x4+84 = 100.

log

-xy + y2

Long 10.

log (x+y)==log!

(4)11x-10-4

代入(3) 7(16-4)-154(10-4)+89"=100"

1*>A,

分解

700-1404+78-150y+15 y2+8y2 = 100-

308-2908+600=0

10(4-3) (34-20)

代入15 得 x=7 或

但因 x=2, y=2° At 12 x-y5q u * *

dog (X~y) &* th

(f) an-b

(解)各取对数

2+ log

.3010. -0.631

(ü) ž 2×3*-1=0, §] =* - 0.5

取対象

log z

X log 3 == logo.

you log 0.5 I6990

0.4771

bútice the last column.

Examples (11) Prove sách of the following 18

contradiction.

Truth table of the prop

(A—98) A (A_An__B)

TRAFF 1 TT

6. G is a necessary and sufficient condition

for Pa

In other words, if P and Q Pare both true, then P and C are equivalent; and if Pisa tautology, then F and Gare

equivalent.

Example:

(1) Frove that A5 and b

equivalent,

To prove A→→→E

Lautology. Truth table {a-

13 C

Xof logb

(x+1)2loga - (X-1) log b

(loga-logb]x2+ 2 (loga+ logb) x + (loga-logb)=0

(未完轉入弟国新三

TFF TT

Since the biconditional is a tautology, the

two statements

q and

are equivalent;

(4) Frove that

Band

AV Bare

«quivalent,

Truth table-

A

Si near the truth wulus of the proposition is false for all possible combinations of the individual statements, it is a contradiction.

(2)SEMA V ~ B) AVAMA B)

Truth table

~BA~ (AAB)

FET

TT

F TT. #

T

Hence the purposition ka contradictiva

Relations of composite propositions

(ay Equivalen

sva tements; or propositions) Two propositions P(p,q,r,...) and 3(5,9,7,...), are said to be logically equivalent if their truth tables are identical, The equivalence of two propositions

•P(p,q,r,...) H(p,q,r,...) is designated by

·F= 6, 0 or

P

Equivalence may be stated in variety of

Maya as follows:

1. Fimplies, and conversely.

2. If then G, and converse.

3. Fif and only if Q.

66

if and only if F.

5 F is a necessary and sufficient condition

for

Therefore,

(2) Prove that B-

equivalent.

Truth table

(BA)<

T.T

F

TT

T

FF

TF

Since the biconditional is a tautology, BANĀTNĒ,

Kemark: In statement logic, AB is called

a conditional, BA is called the converse,

B the inverse, and BA the contrapositive. Hence a given conditional and its contrapositive are equivalent; and its inverse and converse are equivalent.

(3) Prove that the following statements are

equivalentiran

1. Politicians are crooks or my name isn

Smith.

2. If Politicians are not crooks, then my

name isn't Smith

The symbolic form of the two statements are:

~{~p) yuq

ون من

Truth table

TTT TT F

Since the truth tables Of A ----- Band AVB are the same, then A→→→→→ B ⇒À V B.

Work for the week

1. Write the following in symbolic forms

(a) The necessary and sufficient condition for

3x = 18: 15 x = 6.

(b) A is a subset of 3 if and only if ▲ (1) BỊ • p. (c) Jack goes to swim when it is summer. Whenever

it is summer, Helen stays at home

It is not true that he is bright and diligent" and "he is not bright or he is not diligent are equivalent.

Prove each of the following is a tautology. (a) [(A———~~8) ^^ (B →→→→C)] →→→→ (A (b)~(~AA B)- > (AV ~ B)

C) is a tautology.

3. Prove each of the following pairs of statements are

equivalent.

(a) If n is a multiple of 6, then it is a multiple

of 3.

If is not a multiple of 3, then it is not a multiple of 6.

6) If n is even, it is divisible by 2,

`n la not even or n is divisible by

Prove the validity of the following.

If James plays football, then Feter plays piano.

If Peter plays piano, Lily sings. Therefore if James plays football, then Lily sings.