Finitely Generated Abelian Groups
... Corollary: (FERMAT) If p is prime then ap ≡ a (mod p). Proof: If p divides a then LHS = RHS = 0. Otherwise, by Euler’s theorem ap−1 ≡ 1(mod p). Euler's theorem can be used to calculate the remainders of certain very large numbers. Example 12: What is the remainder on dividing 51000 by 42? Solution: ...
... Corollary: (FERMAT) If p is prime then ap ≡ a (mod p). Proof: If p divides a then LHS = RHS = 0. Otherwise, by Euler’s theorem ap−1 ≡ 1(mod p). Euler's theorem can be used to calculate the remainders of certain very large numbers. Example 12: What is the remainder on dividing 51000 by 42? Solution: ...
Total interval numbers of complete r
... play important roles in many applications, see [2]. More generally, we allow a representation f to assign each vertex a union of intervals on the real line. In this case, f is called a multiple-interval representation of the intersection graph of this family of sets. Let #f(v) denote the minimum num ...
... play important roles in many applications, see [2]. More generally, we allow a representation f to assign each vertex a union of intervals on the real line. In this case, f is called a multiple-interval representation of the intersection graph of this family of sets. Let #f(v) denote the minimum num ...
Extracting Proofs from Tabled Proof Search
... of these two strategies is not difficult to establish and it follows the work presented in [8]. Strategy III is sound only when the tabled entries are co-inductive predicates: furthermore, a proof certificate can always be constructed and it will be essentially a post-fixed point found within the ta ...
... of these two strategies is not difficult to establish and it follows the work presented in [8]. Strategy III is sound only when the tabled entries are co-inductive predicates: furthermore, a proof certificate can always be constructed and it will be essentially a post-fixed point found within the ta ...
Cardinality, countable and uncountable sets
... With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify their elements can be placed in pairwise correspondence; that is, that there is a bijection between them. It is then natural to generalize this to infi ...
... With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify their elements can be placed in pairwise correspondence; that is, that there is a bijection between them. It is then natural to generalize this to infi ...
Prime numbers
... Again, no one knows for sure if the Goldbach conjecture is true. There is evidence to think that it is probably true: not only has it been checked for lots and lots of values of N (up to 101 8), but the number of different ways to express N as the sum of two primes increases (albeit unevenly!) as N ...
... Again, no one knows for sure if the Goldbach conjecture is true. There is evidence to think that it is probably true: not only has it been checked for lots and lots of values of N (up to 101 8), but the number of different ways to express N as the sum of two primes increases (albeit unevenly!) as N ...
Lecture notes, sections 2.5 to 2.7
... Let us now the induction proof. The statement S(n) is the statement of the theorem. Base step: The statement S(1) is true, because a set A with one element has exactly two subsets: the empty set and the set A itself. Induction step: Assume that S(n) is true. This means that a set with n elements has ...
... Let us now the induction proof. The statement S(n) is the statement of the theorem. Base step: The statement S(1) is true, because a set A with one element has exactly two subsets: the empty set and the set A itself. Induction step: Assume that S(n) is true. This means that a set with n elements has ...
Induction and Recursion - Bryn Mawr Computer Science
... In the induction step of a proof by mathematical induction, we prove that a natural number has some property based on the assumption that the previous number has the same property. In some cases this assumption isn’t strong enough to make the proof work, and we need to assume that all smaller natura ...
... In the induction step of a proof by mathematical induction, we prove that a natural number has some property based on the assumption that the previous number has the same property. In some cases this assumption isn’t strong enough to make the proof work, and we need to assume that all smaller natura ...
Engaging Students in Proof and Reasoning in High School Non
... help#the#teacher#identify#the#mathematics#available#for#exploration#during#the#lesson(s)#to#follow.#Since# student#thinking#matters#most,#in#the#Arkansas#professional#development#courses##the#beginning#of#most# classes# will# involve# just# making# sense# of# and# deepening# our# understanding# of# ...
... help#the#teacher#identify#the#mathematics#available#for#exploration#during#the#lesson(s)#to#follow.#Since# student#thinking#matters#most,#in#the#Arkansas#professional#development#courses##the#beginning#of#most# classes# will# involve# just# making# sense# of# and# deepening# our# understanding# of# ...
Induction
... L-shapes subject to conditions 1′ and 2′ . A 1 × 1 grid consists of only one square. This square is s, and we can leave it untiled. Since there are no other squares in the grid to tile, this gives us a valid tiling of a 1 × 1 grid. Now we prove the inductive step using a direct proof. Let n ∈ N and ...
... L-shapes subject to conditions 1′ and 2′ . A 1 × 1 grid consists of only one square. This square is s, and we can leave it untiled. Since there are no other squares in the grid to tile, this gives us a valid tiling of a 1 × 1 grid. Now we prove the inductive step using a direct proof. Let n ∈ N and ...
Theorem
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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.