CS103X: Discrete Structures Homework Assignment 2: Solutions
... “Euclids Algorithm finds the correct GCD of a and b in a finite number of steps, for all 0 ≤ a, b ≤ n (a and b not both 0).” We will prove P (n) holds for all positive integers n by induction. We assume a ≥ b: if not, the first couple of steps of the algorithm will take their absolute values and swa ...
... “Euclids Algorithm finds the correct GCD of a and b in a finite number of steps, for all 0 ≤ a, b ≤ n (a and b not both 0).” We will prove P (n) holds for all positive integers n by induction. We assume a ≥ b: if not, the first couple of steps of the algorithm will take their absolute values and swa ...
Truth-Functional Logic
... Parentheses and the Method of Induction A statement like A ∧ B ∨ C is ambiguous: is this the conjunction of A and B ∨ C, or is this a disjunction of A ∧ B and C? We will require that our statements are not ambiguous, and we will use parentheses to do so. So, if this statement was meant to be a conj ...
... Parentheses and the Method of Induction A statement like A ∧ B ∨ C is ambiguous: is this the conjunction of A and B ∨ C, or is this a disjunction of A ∧ B and C? We will require that our statements are not ambiguous, and we will use parentheses to do so. So, if this statement was meant to be a conj ...
Definable maximal cofinitary groups.
... Proof. Let {Dn }n∈ω be an enumeration of all dense open subsets of Qα from M and let {in }n∈ω be an increasing sequence which is cofinal in i. Inductively, construct a sequence hq̄n in∈ω ⊆ M ∩ Qα such that q̄0 = p̄, and (1) for every n ∈ ω, q̄n+1 ≤ q̄n , fin(qn ) = fin(p̄); (2) if inf(qn ) = hdnk , ...
... Proof. Let {Dn }n∈ω be an enumeration of all dense open subsets of Qα from M and let {in }n∈ω be an increasing sequence which is cofinal in i. Inductively, construct a sequence hq̄n in∈ω ⊆ M ∩ Qα such that q̄0 = p̄, and (1) for every n ∈ ω, q̄n+1 ≤ q̄n , fin(qn ) = fin(p̄); (2) if inf(qn ) = hdnk , ...
prime numbers and encryption
... Encryption is the process oftransforming information to make it unreadable to anyone except those possessing special knowledge, usuaiiy referred to as a key. Modem encryption algorithms depends heavily on number theory,with primality testing, factoring, discrete logarithms and elliptic curves being ...
... Encryption is the process oftransforming information to make it unreadable to anyone except those possessing special knowledge, usuaiiy referred to as a key. Modem encryption algorithms depends heavily on number theory,with primality testing, factoring, discrete logarithms and elliptic curves being ...
DMT irm 3 - Information Age Publishing
... does not include experience with reading and writing careful mathematical arguments. Since we must provide middle-grades teachers with not only the specific knowledge they will teach but also the mathematical context for that knowledge, experience with reading and writing proofs is an essential part ...
... does not include experience with reading and writing careful mathematical arguments. Since we must provide middle-grades teachers with not only the specific knowledge they will teach but also the mathematical context for that knowledge, experience with reading and writing proofs is an essential part ...
Abelian and non-Abelian numbers via 3D Origami
... Theory (see, for example, [C04, Corollary 10.1.7]) establishes that the set C of points constructed using ruler and compass (RC, for short) with initial set {0, 1} is the smallest subfield of C which is closed under the operation of taking square roots. Thus, constructions that involve solving cubic ...
... Theory (see, for example, [C04, Corollary 10.1.7]) establishes that the set C of points constructed using ruler and compass (RC, for short) with initial set {0, 1} is the smallest subfield of C which is closed under the operation of taking square roots. Thus, constructions that involve solving cubic ...
Speaking Logic - SRI International
... Interpretation ascribes an intended sense to these assertions by fixing the meaning of certain symbols, e.g., the logical connectives, equality, and delimiting the variation in the meanings of other symbols, e.g., variables, functions, and predicates. An assertion is valid if it holds in all interpr ...
... Interpretation ascribes an intended sense to these assertions by fixing the meaning of certain symbols, e.g., the logical connectives, equality, and delimiting the variation in the meanings of other symbols, e.g., variables, functions, and predicates. An assertion is valid if it holds in all interpr ...
The Hebrew Mathematical Tradition
... explains in the third chapter of his Sefer ha-Shem, the number one is “the cause of all number, but is not itself a number.” Ten resembles one, since it is both the end of the units and the beginning of the “tens”. Moreover, 5, 6, and 10 all share this property in common: all powers of these three n ...
... explains in the third chapter of his Sefer ha-Shem, the number one is “the cause of all number, but is not itself a number.” Ten resembles one, since it is both the end of the units and the beginning of the “tens”. Moreover, 5, 6, and 10 all share this property in common: all powers of these three n ...
Notes on Quadratic Extension Fields
... In this section we are going to show that the answer is NO. To see what’s at stake algebraically, let’s suppose the circle we’re given has radius 1,√so its area is π. Then the square to be constructed must have edge length π. Since a number is constructible if and only if its square is constructible ...
... In this section we are going to show that the answer is NO. To see what’s at stake algebraically, let’s suppose the circle we’re given has radius 1,√so its area is π. Then the square to be constructed must have edge length π. Since a number is constructible if and only if its square is constructible ...
On the Complexity of the Equational Theory of Relational Action
... All relational algebras and lattices, mentioned above, are *-continuous. Consequently, Eq(KA)⊆Eq(KA*)⊆Eq(RKA), and similar inclusions are true for classes KL, KL*, RKL, classes ACTA, ACTA*, RACTA, and classes ACTL, ACTL*, RACTL. It is known that Eq(KA)=Eq(KA*)=Eq(RKA) (this follows from the Kozen co ...
... All relational algebras and lattices, mentioned above, are *-continuous. Consequently, Eq(KA)⊆Eq(KA*)⊆Eq(RKA), and similar inclusions are true for classes KL, KL*, RKL, classes ACTA, ACTA*, RACTA, and classes ACTL, ACTL*, RACTL. It is known that Eq(KA)=Eq(KA*)=Eq(RKA) (this follows from the Kozen co ...
![arXiv:math/0608068v1 [math.NT] 2 Aug 2006](http://s1.studyres.com/store/data/016444901_1-15a2bfb469428d063b2c2745e8d56343-300x300.png)
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.