1972-08-08 — Page 23

★致保单 頁三第張六第日九廿月六年子壬瑟夏 WAH KIU YAT PO

報日僑華

中學同學暑期後個嵩欄

陳道英文書院主编

現代數學

Modern Mathematics (2)

Logical reasoning 、

A statement p is said to

imply a statement g (written as P⇒q) if p⇒ is a tautology. (A tautology has truth value wTh for all possible truth values of its components).

An argument is an assertion. that a given set of statements $1, S2 •****, S. called premises yields (has as a consequence) another statement S called the conclusion. Such an argument wil be denoted by

Note that an argement is tatement and therefore has a

ruth value. If an

a

rue, it is called a

trgument; if an argument: it is called a fallacy..

false,

Way

to test the validity of an argument is to check whether of aot the conjunction of the premises implies the conclusion.

Example (1)

If the U.5. government is democratic, then its citizene have the right to vote.

Its citizens have the right to vote.

The U.5. government is. democratic.

In symbolic form, the reasoning

Since (pa) Aq=P; tnerefore the argument is not valid. However, the conclusion is true by itself.

Example(2)

If a country is democratic. the chief executive is directiv elected by the people.

V The British prime ministe che chief executive, is not. directly elected by people.

Britain is not democratic. In symbolic form, the argument.

The argument is valid since 「(p→a)A (~a)⇒ (~p) but the "conclusion is false by itself.

Therefore we see that the truth of the conclusion is irrelevant as far as the test on. the validity of the argument goes. A true conclusion is neither necessary nor sufficient for the validity of the argument.

Introduction to Set Theory

In discussing any branch of mathematics, it is helpful to

use the notation and terminology.

of set theory. This subject, which

was developed by Bocle and Cantor (George Boole was an English

二期星

Mathematician and logician. His book, An Investigation of the. Laws of Thought, published in 1854, marked the creation of the first: workable system of symbolic logic. George F.L.P. Cantor and his school created the modern. theory of seta during the period (1874–1895) in the latter part of

the 19th century, has had a profound influence on the development of mathematics in the 20th century. It has unified many seemingly disconnected ideas and has helped to reduce many: mathematical concepts to their logical foundations in an elegant and systemic way. A rigorous and formal treatment of the theory of sets would be something-beyond. our scope here. Fortunately, the basic notions 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..

犟

In mathematics, the word "set" is used to represent a✨ collection of objects viewed as a single entity. The collections called to mind by such nouns as "flock", "tribeft, "crowd" "team" are all examples of sets, The individual objects in the" collection are called the elements or neubers of the set,' and they are said to belong to or to be contained in the set, The

aet, in turn, is said to contain or be composed of its elements. Notations for designating sets

Sets usually are denoted b. capital letters: A,B,C,...f elements are designated by lower- case letters: a,b,c,... We us the notation xes to mean that:

" is an element of S" or "x

belong to S". If x does not belong. to. S. we writexes.

There are two methods of describing & Set

日八月八年二七九一蕃公年一十六國民華中

(1)Listing method

We display the elemènta

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 odd integers is displayed as 1,3,5,...} the three dots

mean "and so on".

(2) Property defining method We use the notation

(x:p(x)}where-p(x) is a

statement describing the property of an element x of the set and ":" means such that.

The first basic concept that relates one set to ano the Ls equality of sets: Definition of set equality

Two sets A and Bare said to be equal if they consist of exactly the same elements, in which case we write A=B If one of the gets contains an element not in the other, we say the set are unequal and we write A B.

xample (1)

According to definition, the two sets 1,3,5,7) and (1,5,7,3} are equal since they both consist of the four integers 1,3,5, and 7 Thus we conclude that a set does not change if its elements are rearranged. Example (2)

The sets 1,3,5,7}and {1.1, 3.3.5.5.7.7} are equal even though, in the second set, each. of the elements 1,3,5 and 7 is listed twice. Both sets contain the four elements 1,3,5,7 and no others; therefore the definition requires that they are equal. So we conclude that a set does not change if its elements are repeated ..

|

開常口笑

邊理

主諧全集

演星法中

長怒到保

放心證常惹

!花睇!笑一

5 C

CRACY

Boys

猷算參榮司公代百

映献期下

片距作動笑惹嵇滑

0 更之趣諧。梯突稽滑

域東華總 多 盛

國華

華利城頓統

!請統府闔

睇你保·極新而多不笑 過未証 到且·詆料

!擇選佳最的樂娛期暑

推優抵空本 出選港運片

壓四映在 倒週連巴 一賣滿黎 切座十献

㶅巴娟泰

!會從銀鍊追快拍昇 燕拉端伯比天能

普慶·南華·珠江

南洋·銀

見未兼頭逐艇攝機直

經相巴提学学报

常滿

兵

域多利

肉

利華城統口

【華盛頓

乾命搏艇快餐 滿塲塲 美快匪

激刺對絕 天二十第

·色司公枭:

献報

1972

今天五場

連麥提士利阿!”

每一寸膠片都是真實的報告!

銀都 四場

珠江 南洋

PUPPET ON

A CHAIN

賞飲套圖

版出報日僑華

美華永

“豆彩七門打術武

珠明運海

全部眞實紀錄,鏡頭使人難忘

非人生活實錄

了南日今

王

獨臂

牛點

剝削 勞役。迫害。屠殺

亭五本每

獎大別線或影實城廣匯榮 地天

三角四元二 ̇於學特區

| 七大和 十堰場夜午3

昆士

英

天然七彩鬍 人類在飢餓中掙扎

亞·非·拉人民的痛苦生活!

五洲人民的血淚史

最合各界參攷

讀讀日錄 2年 香港年鑑 每册十五元

最華

食日