Arithmetic Sequences 4.6
... Writing Arithmetic Sequences as Functions Because consecutive terms of an arithmetic sequence have a common difference, the sequence has a constant rate of change. So, the points represented by any arithmetic sequence lie on a line. You can use the first term and the common difference to write a li ...
... Writing Arithmetic Sequences as Functions Because consecutive terms of an arithmetic sequence have a common difference, the sequence has a constant rate of change. So, the points represented by any arithmetic sequence lie on a line. You can use the first term and the common difference to write a li ...
The Dedekind Reals in Abstract Stone Duality
... Since ASD is formulated in a type-theoretical fashion, with absolutely no recourse to set theory, it is intrinsically a computable theory. The familiar arithmetical operations +, − are × are, of course, computable algebraic structure on R, as are division and the (strict) relations <, > and 6= when ...
... Since ASD is formulated in a type-theoretical fashion, with absolutely no recourse to set theory, it is intrinsically a computable theory. The familiar arithmetical operations +, − are × are, of course, computable algebraic structure on R, as are division and the (strict) relations <, > and 6= when ...
No Matter How You Slice It. The Binomial Theorem and - Beck-Shop
... Proof. The right-hand side is, by definition, the number of k + 1-element subsets of [n + 1]. Such a subset S either contains n + 1, or it does not. If it does, then the rest of S is a k-element subset of [n], and these are enumerated by the first member of the left-hand side. If it does not, then S ...
... Proof. The right-hand side is, by definition, the number of k + 1-element subsets of [n + 1]. Such a subset S either contains n + 1, or it does not. If it does, then the rest of S is a k-element subset of [n], and these are enumerated by the first member of the left-hand side. If it does not, then S ...
Default reasoning using classical logic
... In the sequel to this section we will formally justify the translations illustrated above, present the general algorithms, and give more examples. The rest of the paper is organized as follows: After introducing some preliminary de nitions in Section 2, we provide in Section 3 the concept of a mode ...
... In the sequel to this section we will formally justify the translations illustrated above, present the general algorithms, and give more examples. The rest of the paper is organized as follows: After introducing some preliminary de nitions in Section 2, we provide in Section 3 the concept of a mode ...
Chapter 8.1 – 8.5 - MIT OpenCourseWare
... everyone should learn at least a little number theory. In Section 5.4.4, we formalized a state machine for the Die Hard jug-filling problem using 3 and 5 gallon jugs, and also with 3 and 9 gallon jugs, and came to different conclusions about bomb explosions. What’s going on in general? For example, ...
... everyone should learn at least a little number theory. In Section 5.4.4, we formalized a state machine for the Die Hard jug-filling problem using 3 and 5 gallon jugs, and also with 3 and 9 gallon jugs, and came to different conclusions about bomb explosions. What’s going on in general? For example, ...
Carnap and Quine on the analytic-synthetic - Philsci
... Carnap is that Carnap’s notion of analyticity may be too narrow. I will conclude, pace Quine and Carnap, that a broad notion of analyticity may be philosophically useful. In the next section I start by clarifying the analytic-synthetic distinction to be employed in the historical analysis, and to el ...
... Carnap is that Carnap’s notion of analyticity may be too narrow. I will conclude, pace Quine and Carnap, that a broad notion of analyticity may be philosophically useful. In the next section I start by clarifying the analytic-synthetic distinction to be employed in the historical analysis, and to el ...
Lecture Notes for MA 132 Foundations
... • If n0 is prime, then since n0 divides itself, it is divisible by a prime. So it is both divisible by a prime (itself, in this case) and not divisible by a prime (because it is in S). • If n0 is not prime, then we can write it as a product, n0 = n1 n2 , with both n1 and n2 smaller than n0 and bigg ...
... • If n0 is prime, then since n0 divides itself, it is divisible by a prime. So it is both divisible by a prime (itself, in this case) and not divisible by a prime (because it is in S). • If n0 is not prime, then we can write it as a product, n0 = n1 n2 , with both n1 and n2 smaller than n0 and bigg ...
Section 1.4 Proving Conjectures: Deductive Reasoning
... Conjecture: If you multiply two even integers then the product will be even. Let 2m = one even integer Let 2n = a second even integer The product = 2m X 2n = 4mn = 2(2mn) ...
... Conjecture: If you multiply two even integers then the product will be even. Let 2m = one even integer Let 2n = a second even integer The product = 2m X 2n = 4mn = 2(2mn) ...
On integers with many small prime factors
... This would be possible, if Hall’s conjecture [4] on the solutions of y2 = x3+k were proved. (M. Hall Jr. has conjectured that if x, y are integers with x3 ~ y2, then |x3-y2| > |x| for x > xo . ) 6. The following theorem has direct consequences for the number theoretic function f3(n) introd ...
... This would be possible, if Hall’s conjecture [4] on the solutions of y2 = x3+k were proved. (M. Hall Jr. has conjectured that if x, y are integers with x3 ~ y2, then |x3-y2| > |x| for x > xo . ) 6. The following theorem has direct consequences for the number theoretic function f3(n) introd ...
full text (.pdf)
... with every inductive proof that the proof is a valid application of the induction principle. We emphasize that we are not claiming to introduce any new coinductive proof principles. The foundations of coinduction underlying our approach are well known. Rather, our purpose is only to present an infor ...
... with every inductive proof that the proof is a valid application of the induction principle. We emphasize that we are not claiming to introduce any new coinductive proof principles. The foundations of coinduction underlying our approach are well known. Rather, our purpose is only to present an infor ...
Test - Mu Alpha Theta
... set which is the collection of sets that do not contain its own members. If every set is a subset of itself, than this proves contradictory. What is the name of this idea? (a) Cantor's paradox (d) Empty Set contradiction (b) Russell's paradox (e) None of the Above (c) Unit Set theory 18. If Set A = ...
... set which is the collection of sets that do not contain its own members. If every set is a subset of itself, than this proves contradictory. What is the name of this idea? (a) Cantor's paradox (d) Empty Set contradiction (b) Russell's paradox (e) None of the Above (c) Unit Set theory 18. If Set A = ...