Conjunctions In ordinary speech, we frequently join two statements by the word and. Several pressures made foundational issues urgent in the 19th century. Obviously, each of the following sentences is a statement: Geography is a science. Two sets A and B are equal if and only if A is a subset of B and B is a subset of A. In elementary geometry, concepts such as points and lines are undefined; let us therefore accept the notion of a set as undefined.
Hilbert sah sich im Paradies, aus dem man sich nicht mehr vertreiben ließe, Poincaré diagnostizierte eine Krankheit, von der die Mathematik genesen werde. Instead we shall deal with the statements themselves as entireties, and study the modes of compounding them into further statements. T r u t h functional compositions. T h e English or, the French ou and the German oder all admit the two interpretations in common usage. A conditional has two components: the first is called the hypothesis and the second is called the consequent of the conditional. Negations We say that the disjunction and the conjunction are binary compositions because each of them combines two statements into a new statement. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory.
Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Sentence 1 is intended to mean that the city in question, which has a population of nearly four million, is populous, while sentence 2 is intended to mean that the name of the city in question has two syllables. As a law, the axiom of extension has a prohibitive aspect which ensures that the symbol e is 'well-behaved'. Obviously this situation is unsatisfactory, and therefore it is desirable to have some way of distinguishing them.
For B a Ay we sometimes say that B is included in A as a subset or A includes B as a subset. Statement calculus deals only with truth functional compositions. This book, therefore, is intended for schools as well as for universities. Ordinarily, a statement such as 1 is accepted as true if both of its components are true; otherwise it is considered false. Now consider problem 2, which has many applications in mathematics. This and similar problems will be discussed more thoroughly in the next few sections. T h e book begins with a chapter on statement calculus—a brief introduction to the language of logic as used in mathematics.
It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. In mathematical logic, the negation of a statement is formed by prefixing to the statement the tilde ~ which is conveniently read as not. The statement 1 If it rains, then John will drive Jack home or Jack will stay with John. What is the analogous situation in elementary geometry? T h i s , however, will b e done in Section 6 F of this book. A statement formula, which represents a truth functional composition, can also be regarded as a truth function.
In writing Part I, which is a revised version of the offset edition published in 1964, we have had in mind the I long Kong University Advanced Level Examination requirements. P o w e r sets In studying a given set Af we often have to consider its subsets. We now consider sets whose elements are elements of another set. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Let us denote these three statements by Xf Y and Z respectively. Thus each truth functional composition can be described by a truth table.
He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The pattern of this proof is similar to that of proving an algebraic identity. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. The conclusion is preceded by a sequence of equivalences. The title of this section may suggest a hair-splitting pedantry, but before we form such a judgement, let us consider the following two sentences: 1 2 Hong Kong is populous. Taking the absolute complements of both sides in E and applying 2. We have now reached the point where we can treat the statements in the same way as numerals are treated in algebra.
T h e usual way to express a negation is to insert the word not into a sentence or to withdraw it from a sentence as the case may be. There is a chapter on statement calculus, followed by eight chapters on set theory. The symbols V and 3. Thus instead of writing correctly Sec. These examples also show that the truth value of the negation of a statement is just the opposite of that of the statement. In Latin, the words aut and vel correspond to the exclusive and the inclusive or respectively.