Set Theory for Computer Science (pdf )
... sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathematicians were used to thin ...
... sets as completed objects in their own right. Mathematicians were familiar with properties such as being a natural number, or being irrational, but it was rare to think of say the collection of rational numbers as itself an object. (There were exceptions. From Euclid mathematicians were used to thin ...
propositions and connectives propositions and connectives
... two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ...
... two-valued logic – every sentence is either true or false some sentences are minimal – no proper part which is also a sentence others – can be taken apart into smaller parts we can build larger sentences from smaller ones by using connectives ...
Version 1.0 of the Math 135 course notes - CEMC
... Showing Two Sets Are Equal . . . . . . . . . . . . . 8.3.1 Converse of an Implication . . . . . . . . . . . 8.3.2 If and Only If Statements . . . . . . . . . . . 8.3.3 Set Equality and If and Only If Statements . ...
... Showing Two Sets Are Equal . . . . . . . . . . . . . 8.3.1 Converse of an Implication . . . . . . . . . . . 8.3.2 If and Only If Statements . . . . . . . . . . . 8.3.3 Set Equality and If and Only If Statements . ...
A Conditional Logical Framework *
... it could be the case that a redex, depending on the result of a communication, can remain stuck until a “good” message arrives from a given channel, firing in that case an appropriate reduction (this is a common situation in many protocols, where “bad” requests are ignored and “good ones” are served ...
... it could be the case that a redex, depending on the result of a communication, can remain stuck until a “good” message arrives from a given channel, firing in that case an appropriate reduction (this is a common situation in many protocols, where “bad” requests are ignored and “good ones” are served ...
p. 1 Math 490 Notes 4 We continue our examination of well
... can neither be proved nor disproved that ℵ1 = |R|. The assertion that ℵ1 = |R| is called the Continuum Hypothesis. This hypothesis, like the Axiom of Choice, can be accepted or rejected as an axiom of set theory. The Generalized Continuum Hypothesis is the assertion that for any infinite cardinal α, ...
... can neither be proved nor disproved that ℵ1 = |R|. The assertion that ℵ1 = |R| is called the Continuum Hypothesis. This hypothesis, like the Axiom of Choice, can be accepted or rejected as an axiom of set theory. The Generalized Continuum Hypothesis is the assertion that for any infinite cardinal α, ...
Statement
... An argument will not always be written as a sequence in the way described. For example, the second argument above may be written like this: Socrates is mortal, because Socrates is a man and all men are mortal. But everything mortal will cease to exist, so Socrates will cease to exist. Nevertheless, ...
... An argument will not always be written as a sequence in the way described. For example, the second argument above may be written like this: Socrates is mortal, because Socrates is a man and all men are mortal. But everything mortal will cease to exist, so Socrates will cease to exist. Nevertheless, ...
A rational approach to π
... or any other irrational we are looking at. To that end we introduce the following concept. Definition. The irrationality measure of an irrational number α is defined as the limsup over all qualities of all rational approximations and is denoted by µ (α ). We have taken the limsup in our definition r ...
... or any other irrational we are looking at. To that end we introduce the following concept. Definition. The irrationality measure of an irrational number α is defined as the limsup over all qualities of all rational approximations and is denoted by µ (α ). We have taken the limsup in our definition r ...
The Euclidean Algorithm and Its Consequences
... Because of the Key Fact, there is a procedure7 for finding the gcd through repeated division. The precise description of the procedure is somewhat cryptic, but an example will help to make the meaning clear. The Euclidean Algorithm Start with a nonnegative integer a and a positive integer b. Step 1: ...
... Because of the Key Fact, there is a procedure7 for finding the gcd through repeated division. The precise description of the procedure is somewhat cryptic, but an example will help to make the meaning clear. The Euclidean Algorithm Start with a nonnegative integer a and a positive integer b. Step 1: ...
On Subrecursive Representability of Irrational Numbers Lars Kristiansen
... can we compute a trace function for α subrecursively in the Dedekind cut of α (we assume that α is irrational)? Let us say that we want to determine a value for T(1). Any value closer to α than 1 will do. We ask the Dedekind cut which tells us that 1 < α . Now we know that T(1) should be strictly gr ...
... can we compute a trace function for α subrecursively in the Dedekind cut of α (we assume that α is irrational)? Let us say that we want to determine a value for T(1). Any value closer to α than 1 will do. We ask the Dedekind cut which tells us that 1 < α . Now we know that T(1) should be strictly gr ...
A rational approach to
... rationals approximating suciently well to establish its irrationality. In the long history of this is a recent result indeed. But there is more. In general, irrationality proofs obtained by explicit construction of rational approximations yield more information than just an irrationality proof. ...
... rationals approximating suciently well to establish its irrationality. In the long history of this is a recent result indeed. But there is more. In general, irrationality proofs obtained by explicit construction of rational approximations yield more information than just an irrationality proof. ...
propositional logic extended with a pedagogically useful relevant
... All this will sound familiar to people acquainted with the work of Anderson and Belnap. The language being W 1 , there is no need for index sets; the star will be sufficient to recall whether the hypothesis of the subproof is or is not relevant to the conclusion of the subproof. If it is, an arrow c ...
... All this will sound familiar to people acquainted with the work of Anderson and Belnap. The language being W 1 , there is no need for index sets; the star will be sufficient to recall whether the hypothesis of the subproof is or is not relevant to the conclusion of the subproof. If it is, an arrow c ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.