MATH 311–01 Exam #1 Solutions 1. (7 points) Consider the true
... The converse of this statement is “for any natural number n, if n is a perfect square, then it has an odd number of factors.” This is a true statement. One easy way to see it to be true is that most of the factors of n2 come in pairs: whenever ab = n2 , both a and b are factors. The one exception to ...
... The converse of this statement is “for any natural number n, if n is a perfect square, then it has an odd number of factors.” This is a true statement. One easy way to see it to be true is that most of the factors of n2 come in pairs: whenever ab = n2 , both a and b are factors. The one exception to ...
Theorem (Infinitude of Prime Numbers).
... Theorem: there is an infinite number of prime numbers (The proof of this theorem is attributed to Euclid since it appears in Euclid’s element circa 300 B.C.) Note: This is not a formal proof of the theorem. This is just the general strategy of the proof illustrated with a particular example. There a ...
... Theorem: there is an infinite number of prime numbers (The proof of this theorem is attributed to Euclid since it appears in Euclid’s element circa 300 B.C.) Note: This is not a formal proof of the theorem. This is just the general strategy of the proof illustrated with a particular example. There a ...
1 BASIC COUNTING RULES
... At most 10 computers are in use at one time. Every 5 minutes, some subset of computers requests printers. We want to connect each computer to some of the printers so that we should use as few connections as possible but we should be always sure that a computer will have a printer to use. (At most on ...
... At most 10 computers are in use at one time. Every 5 minutes, some subset of computers requests printers. We want to connect each computer to some of the printers so that we should use as few connections as possible but we should be always sure that a computer will have a printer to use. (At most on ...
valid - Informatik Uni Leipzig
... on F and let w be an arbitrary world in I . If 2ϕ is not true in a world w, then axiom T is true in w. If 2ϕ is true in w, then ϕ is true in all accessible worlds. Since the accessibility relation is reflexive, w is among the accessible worlds, i.e., ϕ is true in w. This means that also in this case ...
... on F and let w be an arbitrary world in I . If 2ϕ is not true in a world w, then axiom T is true in w. If 2ϕ is true in w, then ϕ is true in all accessible worlds. Since the accessibility relation is reflexive, w is among the accessible worlds, i.e., ϕ is true in w. This means that also in this case ...
EGYPTIAN FRACTIONS WITH EACH DENOMINATOR HAVING
... Now we finally return to (1). We proceed by placing many copies of pn+4 Ln+3 (n) (where the middle consists of a long run of terms which differ by pn+4 ), one for each element of Ln+3 (n + 1) which act as translates. We claim that we have every element of Ln+4 (n + 1) which is congruent to r0 (mod p ...
... Now we finally return to (1). We proceed by placing many copies of pn+4 Ln+3 (n) (where the middle consists of a long run of terms which differ by pn+4 ), one for each element of Ln+3 (n + 1) which act as translates. We claim that we have every element of Ln+4 (n + 1) which is congruent to r0 (mod p ...
A COMBINATORIAL PROOF OF A RESULT FROM NUMBER
... (4, 0, 0) (three possible permutations), and (3, 2, 1) (six possible permutations), giving a total of nine solutions. Geometrically, rk (n) counts the number of points with integer coordinates on the kdimensional sphere x21 + x22 + · · · + x2k = n. Similarly, 2k tk (n) counts the number of points wi ...
... (4, 0, 0) (three possible permutations), and (3, 2, 1) (six possible permutations), giving a total of nine solutions. Geometrically, rk (n) counts the number of points with integer coordinates on the kdimensional sphere x21 + x22 + · · · + x2k = n. Similarly, 2k tk (n) counts the number of points wi ...
(n!)+
... “There are infinitely many prime numbers.” Any finite set of numbers must contain a maximal element, so we can prove the theorem if we can just show that there is no largest prime number. i.e., show that for any prime number, there is a larger number that is also prime. More generally: For any numbe ...
... “There are infinitely many prime numbers.” Any finite set of numbers must contain a maximal element, so we can prove the theorem if we can just show that there is no largest prime number. i.e., show that for any prime number, there is a larger number that is also prime. More generally: For any numbe ...
the well-ordering principle - University of Chicago Math
... As an application, we prove: Proposition. Every rational number can be written in lowest terms. That is, every q ∈ Q can be written as q = ab where a and b are integers with no common factor greater than one. Proof. Let A be the set of values of |b| for all fractions ab which cannot be written in lo ...
... As an application, we prove: Proposition. Every rational number can be written in lowest terms. That is, every q ∈ Q can be written as q = ab where a and b are integers with no common factor greater than one. Proof. Let A be the set of values of |b| for all fractions ab which cannot be written in lo ...
Introduction to the Theory of Computation
... Jack sees Jill, who has just come in from outdoors Proof by Induction dry. Jack knows that it is not raining. For any n, there exist n consecutive composite integers. His proof: if it were raining (assuming the statement Proof: here Jill are the n be consecutive composite integers is false), woud we ...
... Jack sees Jill, who has just come in from outdoors Proof by Induction dry. Jack knows that it is not raining. For any n, there exist n consecutive composite integers. His proof: if it were raining (assuming the statement Proof: here Jill are the n be consecutive composite integers is false), woud we ...
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.