Lecture 3.1
... the relation orders only some elements not all E.g. “less than equal to” () on complex numbers; Consider (2 + 4i) and (3 + 2i) ...
... the relation orders only some elements not all E.g. “less than equal to” () on complex numbers; Consider (2 + 4i) and (3 + 2i) ...
Lecture 3
... the relation orders only some elements not all E.g. “less than equal to” () on complex numbers; Consider (2 + 4i) and (3 + 2i) ...
... the relation orders only some elements not all E.g. “less than equal to” () on complex numbers; Consider (2 + 4i) and (3 + 2i) ...
characterization of prime numbers by
... Thus, x → y differs from x → y in that x → y does not always take the k designated value n − 1 when x < y and if x = y then x → y = n − 1 only when x ∈ {0, 1, n − 1}. Logic Kn is defined in analogy with Ln , and Kn is the set of all matrix functions from Kn . To prove the theorems the following two ...
... Thus, x → y differs from x → y in that x → y does not always take the k designated value n − 1 when x < y and if x = y then x → y = n − 1 only when x ∈ {0, 1, n − 1}. Logic Kn is defined in analogy with Ln , and Kn is the set of all matrix functions from Kn . To prove the theorems the following two ...
Full text
... as o(n) = 2n9 a(ft) = kn (k _> 2, an integer), or a(ft) = 2n-f 1, respectively, where o(n) is the sum of the positive divisors of ft. No odd multiperfect numbers are known. In many papers concerned with odd perfect numbers (summarized in McDaniel & Hagis [5]), values have been obtained which cannot ...
... as o(n) = 2n9 a(ft) = kn (k _> 2, an integer), or a(ft) = 2n-f 1, respectively, where o(n) is the sum of the positive divisors of ft. No odd multiperfect numbers are known. In many papers concerned with odd perfect numbers (summarized in McDaniel & Hagis [5]), values have been obtained which cannot ...
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Homework
... Theorem 4: ℤn is an integral domain if and only if n is a prime number. Suppose that n is prime. To show that ℤp is an integral domain let a be an element of ℕ = {1, 2, 3, … p-1} and that a has no inverse in ℤp. Then none of the p numbers 0a, 1a, 2a, … (p-1)a can be equal to 1, so this list must co ...
... Theorem 4: ℤn is an integral domain if and only if n is a prime number. Suppose that n is prime. To show that ℤp is an integral domain let a be an element of ℕ = {1, 2, 3, … p-1} and that a has no inverse in ℤp. Then none of the p numbers 0a, 1a, 2a, … (p-1)a can be equal to 1, so this list must co ...
Simple multiplicative proof nets with units
... conclusion of a ⊥-link) works. Worse, this new jump is by no means natural (if A is B ⊗ C, the new jump can either be B or C), which is quite unpleasant. As far as we know, the only solution consists in declaring that jumps are not part of the proof-net, but rather some control structure. It is then ...
... conclusion of a ⊥-link) works. Worse, this new jump is by no means natural (if A is B ⊗ C, the new jump can either be B or C), which is quite unpleasant. As far as we know, the only solution consists in declaring that jumps are not part of the proof-net, but rather some control structure. It is then ...
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.