What lifts?

As is well known, idempotents in any ring R lift modulo any nil idea I. That is, if a ? R and (a₂-a) ? I there is an i ? I with (a + i)²-- (a + i) = 0. An idempotent is a zero of the polynomial x ²?x ², and a nil element satisfies x ⁿfor some n. Seen this way, lifting occurs in considerable generality We assume R has a unit, and handle the non-unital case at the end of this paper.     

What is continuity?

A continuous process is a person who draws the curve without a break,or,takes place gradually without interruption. Those are a daily language in continuity. However,in calculus,we notice that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. It means a continuous function     

What Is Entitlement?

In his seminal paper, “Content Preservation,” Tyler Burge defends an original account of testimonial knowledge. The originality of the account is due, in part, to the fact that it is cast within a novel epistemic framework. The central feature of that framework is the introduction of the concept of entitlement, which is alleged to be a distinctive type of positive epistemic support or warrant. Entitlement and justification, according to Burge, are sub-species of warrant. Justification is the internalist form of warrant, but entitlement is epistemically externalist. My focus in this paper is Burge’s conception of entitlement, and there are three primary issues that I wish to address. What is the relationship between entitlement and the more traditional concept of justification? In what sense is entitlement epistemically externalist? Has Burge introduced a new epistemic concept or merely coined a new term for a familiar epistemic concept?     

Closed products of sets and?the?axiom of choice

With a slight modification of a previous argument due to Schechter, we show that the Axiom of Choice is equivalent to the following topological statement: ??If a product of a non-empty family of sets is closed in a topological (Tychonoff) product, then at least one of the factors is closed??. We also discuss the case on which one adds the hypothesis that the closed product of sets is a non-empty set.     

What is Inductive Reasoning?

As a child you learned by experimenting with the natural world about you．You learned how to walk．talk,and hold your spoon,all by trial and error．After many tries,you eventually learned how to ride your first bicycle,discovering it was easier to keep your bal- ance if you kept the bike moving．You learned by expe- rience that counterclockwise motion turns on a water faucet and a clockwise motion turns it off．Most of your learning has been by a process called inductive reasoning．     

What Norm of Assertion?

I argue that the debates over which norm constitutes assertion can be abandoned by challenging the three main motivations for a constitutive norm. The first motivation is the alleged analogy between language and games. The second motivation is the intuition that some assertions are worthy of criticism. The third is the discursive responsibilities incurred by asserting. I demonstrate that none of these offer good reasons to believe in a constitutive norm of assertion, as such a norm is understood in the literature. Others who have made similar arguments conclude that assertion does not exist at all—that there is no such thing as assertion. I disagree: we do not have to relinquish the category of assertion just because it is not normatively constituted. There are alternative ways to understand and individuate assertion that do not rely on a constitutive norm.     

What is a Graph?

In the past few years, the subject of graph theory (or network analysis) has come very much to the fore, not only as an important mathematical discipline in its own right, but also as a useful mathematical tool in a wide variety of subjects, ranging from organic chemistry and probability, through operational research and geography, to sociology and linguistics. In this article, we shall present some simple applications of graph theory, chosen in such a way as to be as elementary but as varied as possible. The reader who is interested in pursuing the subject further should consult Reference 1; for a lengthy account of some further applications of graph theory, see Chapter six of Reference 2.     

What Are Equivalent Equations?

Theequationsx =6 38,4 32and 6 0 6 ,90 0 -x =- 31,532areequivalentequationsbecausetheyhavethesamesolu tions .Whenyouwritedownthestepstosolveanequation ,eachstepshouldbeanequationequivalenttotheequationsintheprevioussteps .ExampleMultiplechoice .Whichequationisnotequivalenttotheothers?(a)x +3=8　 (b) 2x =10　 (c) - 1=4 -x　 (d)x =5　 (e)x - 0 .4 =4 .96SolutionEquations (a) ,(b) ,(c)and (d)allhavethesinglesolution 5.Thesolutiontoequation (e)is5.36 .Equatin (e)isnotequivalenttotheothers .Ofalleq…     

What is quasiconvex analysis?

We endeavour to answer the question of the title, or rather the question of how much one can extend convex analysis to a wider framework in which some convexity features remain. Subdifferential calculus and duality are the main directions we consider