![Proof that 2+2=4](http://s1.studyres.com/store/data/000423540_1-7580a2b6b746698407c369845a96d0fc-300x300.png)
Proof that 2+2=4
... This is a contradiction to Fermat’s Little Theorem, so 2 + 2 must not be prime. But 3 is prime. Hence 2 + 2 must not equal 3, and therefore 2 + 2 ≥ 4. It remains to show that 2 + 2 is not greater than 4. To prove this we need the following lemma: Lemma 1. ∀a ∈ Z, if a > 4, ∃b ∈ Z such that b > 0 and ...
... This is a contradiction to Fermat’s Little Theorem, so 2 + 2 must not be prime. But 3 is prime. Hence 2 + 2 must not equal 3, and therefore 2 + 2 ≥ 4. It remains to show that 2 + 2 is not greater than 4. To prove this we need the following lemma: Lemma 1. ∀a ∈ Z, if a > 4, ∃b ∈ Z such that b > 0 and ...
Midterm #3: practice
... (a) Modulo 323, what do we learn from Euler's theorem? (b) Using the Chinese remainder theorem, show that x144 1 (mod 323) for all x coprime to 323. (c) Compare the two results! Bonus: Can you come up with a strengthening of Euler's theorem? ...
... (a) Modulo 323, what do we learn from Euler's theorem? (b) Using the Chinese remainder theorem, show that x144 1 (mod 323) for all x coprime to 323. (c) Compare the two results! Bonus: Can you come up with a strengthening of Euler's theorem? ...
Full text
... 2: Let IT be a partition whose Ferrers graph is embedded in the fourth quadrant. Each node (i, j) of the fourth quadrant which is not in the Ferrers graph of IT is said to possess an anti-hook difference p^ - kj relative to ir, where p^ is the number of nodes on the i t h row of the fourth quadrant ...
... 2: Let IT be a partition whose Ferrers graph is embedded in the fourth quadrant. Each node (i, j) of the fourth quadrant which is not in the Ferrers graph of IT is said to possess an anti-hook difference p^ - kj relative to ir, where p^ is the number of nodes on the i t h row of the fourth quadrant ...
Class notes, rings and modules : some of 23/03/2017 and 04/04/2017
... to be “aligned” with a basis of M in order to compute the quotient. Thus, our goal now is to find such aligned bases for the submodule N from the first picture. Here is an algorithm for doing it. We write the coordinates of the generators of N as rows of a matrix (called relations matrix). In our ex ...
... to be “aligned” with a basis of M in order to compute the quotient. Thus, our goal now is to find such aligned bases for the submodule N from the first picture. Here is an algorithm for doing it. We write the coordinates of the generators of N as rows of a matrix (called relations matrix). In our ex ...
Notes Predicate Logic
... theorems. In this case, the theorem could be written: If x is rational, then x is real. In general, a univeral quantification asserts that if an element is within an understood universe or within a specified domain, then the statement that follows is true. Propositional logic doesn’t work as well wi ...
... theorems. In this case, the theorem could be written: If x is rational, then x is real. In general, a univeral quantification asserts that if an element is within an understood universe or within a specified domain, then the statement that follows is true. Propositional logic doesn’t work as well wi ...
1.4 Proving Conjectures: Deductive Reasoning
... Caribbean. All the following statements about their land areas are true. List the countries in order of increasing size. i. Barbados is smaller than Trinidad and Tobago ii. Bahamas is neither the largest nor the smallest. iii. At least two countries are larger than Trinidad and Tobago. ...
... Caribbean. All the following statements about their land areas are true. List the countries in order of increasing size. i. Barbados is smaller than Trinidad and Tobago ii. Bahamas is neither the largest nor the smallest. iii. At least two countries are larger than Trinidad and Tobago. ...
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.