eprint_4_1049_36.doc
... (c) Distributive law: a(b + c) = ab + ac (d) Additive identity 0 and multiplicative identity 1: a+0=0+a=a ...
... (c) Distributive law: a(b + c) = ab + ac (d) Additive identity 0 and multiplicative identity 1: a+0=0+a=a ...
Lecture 2 - Thursday June 30th
... As an example of induction, we can show 1 + 2 + · · · + k = k(k + 1)/2 for all k ∈ N. Let P (k) be this statement. Then P (1) is the statement 1 = 1(1 + 1)/2 which is true. If P (k) is true, can we show P (k + 1) is also true? Yes, since P (k + 1) is the statement 1 + 2 + · · · + (k + 1) = (k + 1)(k ...
... As an example of induction, we can show 1 + 2 + · · · + k = k(k + 1)/2 for all k ∈ N. Let P (k) be this statement. Then P (1) is the statement 1 = 1(1 + 1)/2 which is true. If P (k) is true, can we show P (k + 1) is also true? Yes, since P (k + 1) is the statement 1 + 2 + · · · + (k + 1) = (k + 1)(k ...
EMBEDDING AN ANALYTIC EQUIVALENCE RELATION IN THE
... In applications to Bayesian decision theory and game theory, it is reasonable to specify each agent’s information as a ∆11 (that is, Borel) equivalence relation, or even as a smooth or closed Borel relation.5 Thus it may be asked: if the graphs of E1 and E2 are in ∆11 or in some smaller class, then ...
... In applications to Bayesian decision theory and game theory, it is reasonable to specify each agent’s information as a ∆11 (that is, Borel) equivalence relation, or even as a smooth or closed Borel relation.5 Thus it may be asked: if the graphs of E1 and E2 are in ∆11 or in some smaller class, then ...
BASIC SET THEORY
... the singleton set containing x . THEOREM: {x } = {y } if and only if x = y . Proof: If {x } = {y }, then: Any member of {y } is also a member of {x }. Any member of {x } is equal to x itself. So any member of {y } is equal to x itself. So {y } is the singleton set {x }. The first several weeks of a ...
... the singleton set containing x . THEOREM: {x } = {y } if and only if x = y . Proof: If {x } = {y }, then: Any member of {y } is also a member of {x }. Any member of {x } is equal to x itself. So any member of {y } is equal to x itself. So {y } is the singleton set {x }. The first several weeks of a ...
A Theory of Theory Formation
... – Tidy definitions up – Repetitions, function conflict, negation conflict ...
... – Tidy definitions up – Repetitions, function conflict, negation conflict ...
A Finite Model Theorem for the Propositional µ-Calculus
... 2. ≤ is well-founded, and there is no infinite set of pairwise ≤-incomparable elements. 3. Every countable sequence x0 , x1 , . . . has xi ≤ xj for some i < j. 4. Every countable sequence x0 , x1 , . . . has a countable monotone subsequence ...
... 2. ≤ is well-founded, and there is no infinite set of pairwise ≤-incomparable elements. 3. Every countable sequence x0 , x1 , . . . has xi ≤ xj for some i < j. 4. Every countable sequence x0 , x1 , . . . has a countable monotone subsequence ...
Infinity + Infinity
... Now, when students discuss infinity, assuming they know no set theory or any of Georg Cantor’s work, they are discussing the cardinality of N, which is defined as |N| = ℵ0 (”alephnaught”). We must consider three concepts before we can make sense of ∞ + ∞ = ℵ0 + ℵ0 . 1) Cantor-Bernstein-Schröeder Th ...
... Now, when students discuss infinity, assuming they know no set theory or any of Georg Cantor’s work, they are discussing the cardinality of N, which is defined as |N| = ℵ0 (”alephnaught”). We must consider three concepts before we can make sense of ∞ + ∞ = ℵ0 + ℵ0 . 1) Cantor-Bernstein-Schröeder Th ...
Number Theory I: Divisibility Divisibility Primes and composite
... Definition: Let n ∈ N with n ≥ 2. Then n is called a prime if its only positive divisors are 1 and n, and n is called composite otherwise. Here is an equivalent form of this definition that is particularly useful for proofs: Definition: An integer n ≥ 2 is composite if it can be written in the form ...
... Definition: Let n ∈ N with n ≥ 2. Then n is called a prime if its only positive divisors are 1 and n, and n is called composite otherwise. Here is an equivalent form of this definition that is particularly useful for proofs: Definition: An integer n ≥ 2 is composite if it can be written in the form ...
1 The Natural Numbers
... approaches the non-rational number 2, a fact well known since antiquity. We want to remedy this deficiency: we want to construct an ordered field F containing the rational numbers, which is “complete” in the following sense: (C1) Every increasing13 bounded14 sequence15 of elements in F converges16 t ...
... approaches the non-rational number 2, a fact well known since antiquity. We want to remedy this deficiency: we want to construct an ordered field F containing the rational numbers, which is “complete” in the following sense: (C1) Every increasing13 bounded14 sequence15 of elements in F converges16 t ...
PDF
... The iterated totient function φk (n) is ak in the recurrence relation a0 = n and ai = φ(ai−1 ) for i > 0, where φ(x) is Euler’s totient function. After enough iterations, the function eventually hits 2 followed by an infinite trail of ones. Ianucci et al define the “class” c of n as the integer such ...
... The iterated totient function φk (n) is ak in the recurrence relation a0 = n and ai = φ(ai−1 ) for i > 0, where φ(x) is Euler’s totient function. After enough iterations, the function eventually hits 2 followed by an infinite trail of ones. Ianucci et al define the “class” c of n as the integer such ...
Separating classes of groups by first–order sentences
... sentences such that ϕ ∈ T ⇔ F (ϕ) ∈ S. We work towards classifying the computational complexity of the theories Th(C), where C is a class from List 1.2. All theories are known to be undecidable, as a consequence of results in [11]. An obvious upper bound for theories of the classes under items (1)–( ...
... sentences such that ϕ ∈ T ⇔ F (ϕ) ∈ S. We work towards classifying the computational complexity of the theories Th(C), where C is a class from List 1.2. All theories are known to be undecidable, as a consequence of results in [11]. An obvious upper bound for theories of the classes under items (1)–( ...
Cardinality, countable and uncountable sets
... consider some unusual subsets of the real line, and it is then natural to wonder if one can give a precise meaning to the intuitive feeling that some infinite sets have “more elements” than other infinite sets (for example, the real line seems to have “more elements” than, say, the rational numbers ...
... consider some unusual subsets of the real line, and it is then natural to wonder if one can give a precise meaning to the intuitive feeling that some infinite sets have “more elements” than other infinite sets (for example, the real line seems to have “more elements” than, say, the rational numbers ...
Solutions to Exercises Chapter 2: On numbers and counting
... of its arguments. Prove that there are 22 different Boolean functions. Why is this the same as the number of families of sets? Let x1 , . . . , xn be the n arguments. There are two choices for the value of each, and hence 2n for the number of different “inputs”. For each input the function takes one ...
... of its arguments. Prove that there are 22 different Boolean functions. Why is this the same as the number of families of sets? Let x1 , . . . , xn be the n arguments. There are two choices for the value of each, and hence 2n for the number of different “inputs”. For each input the function takes one ...