Congruences
... even have the algebraic notation that we take for granted today. Even after the advent of Algebra, it took a couple of hundred years before Gauss in Europe, came up with the notation we use today for congruences. This notation probably opened the floodgates, for in the next couple of hundred years w ...
... even have the algebraic notation that we take for granted today. Even after the advent of Algebra, it took a couple of hundred years before Gauss in Europe, came up with the notation we use today for congruences. This notation probably opened the floodgates, for in the next couple of hundred years w ...
Lecture Notes on Discrete Mathematics
... 1. [Subset of a set] If C is a set such that each element of C is also an element of A, then C is said to be a subset of the set A, denoted C ⊆ A. 2. [Equality of sets] The sets A and B are said to be equal if A ⊆ B and B ⊆ A, denoted A = B. ...
... 1. [Subset of a set] If C is a set such that each element of C is also an element of A, then C is said to be a subset of the set A, denoted C ⊆ A. 2. [Equality of sets] The sets A and B are said to be equal if A ⊆ B and B ⊆ A, denoted A = B. ...
2 Sequences: Convergence and Divergence
... an cannot approach any one specific number a as n grows large. Also, we note that if a is any real number, we can always choose a positive number such that at least one of the inequalities a − < −1 < a + or a − < 1 < a + is false. For example, the choice = |1 − a|/2 if a = 1, and = |1 ...
... an cannot approach any one specific number a as n grows large. Also, we note that if a is any real number, we can always choose a positive number such that at least one of the inequalities a − < −1 < a + or a − < 1 < a + is false. For example, the choice = |1 − a|/2 if a = 1, and = |1 ...
Elementary Number Theory: Primes, Congruences
... our ability to communicate securely. The goal of this book is to bring the reader closer to this world. The reader is strongly encouraged to do every exercise in this book, checking their answers in the back (where many, but not all, solutions are given). Also, throughout the text there, are example ...
... our ability to communicate securely. The goal of this book is to bring the reader closer to this world. The reader is strongly encouraged to do every exercise in this book, checking their answers in the back (where many, but not all, solutions are given). Also, throughout the text there, are example ...
3 Congruence arithmetic
... Note: We will not prove the following elementary facts, but 1·g = g for all g is equivalent to g·1 = g for all g, and g −1 g = 1 is equivalent to gg −1 = 1, so one does not need to check multiplications both ways. (This is of course trivial when G is abelian—our primary case of interest.) Also cond ...
... Note: We will not prove the following elementary facts, but 1·g = g for all g is equivalent to g·1 = g for all g, and g −1 g = 1 is equivalent to gg −1 = 1, so one does not need to check multiplications both ways. (This is of course trivial when G is abelian—our primary case of interest.) Also cond ...
Synopsis of linear associative algebra. A report on its natural
... seems to have been Cayley's3 view. It is in essence the view of most writers on the subject. The other regards the number in a linear algebra as a single entity, and multiplex only in that an equality between two such numbers implies n equalities between certain coordinates or functions of the numbe ...
... seems to have been Cayley's3 view. It is in essence the view of most writers on the subject. The other regards the number in a linear algebra as a single entity, and multiplex only in that an equality between two such numbers implies n equalities between certain coordinates or functions of the numbe ...
Math 25: Solutions to Homework # 4 (4.3 # 10) Find an integer that
... and y are unreadable digits. How much did each chicken cost? ...
... and y are unreadable digits. How much did each chicken cost? ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.