
PPT - UBC Department of CPSC Undergraduates
... • Work forward, playing around with what you can prove from the premises • Work backward, considering what you’d need to reach the conclusion • Play with the form of both premises and conclusions using logical equivalences Finally, disproving something is just proving ...
... • Work forward, playing around with what you can prove from the premises • Work backward, considering what you’d need to reach the conclusion • Play with the form of both premises and conclusions using logical equivalences Finally, disproving something is just proving ...
Fuchsian groups, coverings of Riemann surfaces, subgroup growth
... Theorem 1.7 Let Γ be a Fuchsian group. Then the probability that a random homomorphism in Homtrans (Γ, An ) is an epimorphism tends to 1 as n → ∞. Moreover, this probability is 1 − O(c−n ) for some constant c > 1 depending on Γ. Our proofs show that in Theorems 1.5, 1.6 and 1.7, any constant c satis ...
... Theorem 1.7 Let Γ be a Fuchsian group. Then the probability that a random homomorphism in Homtrans (Γ, An ) is an epimorphism tends to 1 as n → ∞. Moreover, this probability is 1 − O(c−n ) for some constant c > 1 depending on Γ. Our proofs show that in Theorems 1.5, 1.6 and 1.7, any constant c satis ...
Full-Text PDF - EMS Publishing House
... 3. Corollary: . . . If ab = 4 and a = 4 . . . , then also b = 4 . 4. If all prime numbers were of the form 4 , then every number at all should be contained in the form 4 . 5. An arbitrary prime number p being proposed, there always is some number of the form a 2 + b2 + c2 + d 2 which is divisible ...
... 3. Corollary: . . . If ab = 4 and a = 4 . . . , then also b = 4 . 4. If all prime numbers were of the form 4 , then every number at all should be contained in the form 4 . 5. An arbitrary prime number p being proposed, there always is some number of the form a 2 + b2 + c2 + d 2 which is divisible ...
Polynomials with integer values.
... might be ‘God gave us integers and all else is man’s work.’ All of us are familiar already from middle school with the similarities between the set of integers and the set of all polynomials in one variable. A paradigm of this is the Euclidean (division) algorithm. However, it requires an astute obs ...
... might be ‘God gave us integers and all else is man’s work.’ All of us are familiar already from middle school with the similarities between the set of integers and the set of all polynomials in one variable. A paradigm of this is the Euclidean (division) algorithm. However, it requires an astute obs ...
this paper - lume ufrgs
... snffices to show tha.t e~·~ ) :f=. 1 (rnod R), where R > 1 is any factor of n. Using this idea, we study the converse of VVolstenholme's theorem for powers of primes p, by dctcrmining thc valuc of thc binomial cocfficicnt modulo p 3 , p 11 and p 5 . In section 4 we prove that if n is a power of 3, t ...
... snffices to show tha.t e~·~ ) :f=. 1 (rnod R), where R > 1 is any factor of n. Using this idea, we study the converse of VVolstenholme's theorem for powers of primes p, by dctcrmining thc valuc of thc binomial cocfficicnt modulo p 3 , p 11 and p 5 . In section 4 we prove that if n is a power of 3, t ...
Keys GEO SY13-14 Openers 4-3
... For all numbers a & b, if a = b, then b = a. For all numbers a, b & c, if a = b and b = c, then a = c. For all numbers a, b & c, if a = b, then a + c = b + c & a – c = b – c. For all numbers a, b & c, if a = b, then a * c = b * c & a ÷ c = b ÷ c. For all numbers a & b, if a = b, then a may be replac ...
... For all numbers a & b, if a = b, then b = a. For all numbers a, b & c, if a = b and b = c, then a = c. For all numbers a, b & c, if a = b, then a + c = b + c & a – c = b – c. For all numbers a, b & c, if a = b, then a * c = b * c & a ÷ c = b ÷ c. For all numbers a & b, if a = b, then a may be replac ...
PDF
... where the symbols ¬, ∧, and ∨ in Lc are used as abbreviational tools (see the first remark). Then we see that the proposition makes sense. 4. Another way of getting around this issue is to come up with another axiom system for PLc that uses →, ¬, ∧, and ∨ as primitive logical connectives. Such an ax ...
... where the symbols ¬, ∧, and ∨ in Lc are used as abbreviational tools (see the first remark). Then we see that the proposition makes sense. 4. Another way of getting around this issue is to come up with another axiom system for PLc that uses →, ¬, ∧, and ∨ as primitive logical connectives. Such an ax ...
Totient Theorem
... Lemma 1: Each number in the first set must be congruent to one and only one number in the second and each number in the second set must be congruent to one and only one number in the first. This may not be obvious at first but can be proved through three logical steps. (1) Each number in the first s ...
... Lemma 1: Each number in the first set must be congruent to one and only one number in the second and each number in the second set must be congruent to one and only one number in the first. This may not be obvious at first but can be proved through three logical steps. (1) Each number in the first s ...
Problem-Solving Strategies: Research Findings from Mathematics
... by the individual that can be brought to bear on the problem at hand), heuristics, control (global decisions regarding the selection and implementation of resources and strategies) and belief systems (one's perspectives regarding the nature of mathematics and how one goes about working it). This pap ...
... by the individual that can be brought to bear on the problem at hand), heuristics, control (global decisions regarding the selection and implementation of resources and strategies) and belief systems (one's perspectives regarding the nature of mathematics and how one goes about working it). This pap ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... provide cut-free systems for all logics in the cube, they do not provide cut-free systems for all possible combinations of axioms. For example, the logic S5 can be obtained by adding b and 4, or by adding t and 5, to the modal logic K, but a complete cut-free system could only be obtained by adding ...
... provide cut-free systems for all logics in the cube, they do not provide cut-free systems for all possible combinations of axioms. For example, the logic S5 can be obtained by adding b and 4, or by adding t and 5, to the modal logic K, but a complete cut-free system could only be obtained by adding ...
- ScholarWorks@GVSU
... that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.) Use of prior knowledge. This also is ...
... that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.) Use of prior knowledge. This also is ...
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.