Sols - Tufts Math Multi
... Answer. Fix any partition P of [0, 1]. That partition must have only finitely many subintervals of length greater than 1/n... certainly it can’t have more than n of them, because the pieces of a partition are disjoint. So let’s list all the intervals within P in order of size, from biggest to smalle ...
... Answer. Fix any partition P of [0, 1]. That partition must have only finitely many subintervals of length greater than 1/n... certainly it can’t have more than n of them, because the pieces of a partition are disjoint. So let’s list all the intervals within P in order of size, from biggest to smalle ...
N Reals in (0,1)
... Set-up: Can the set of all rational numbers Q can be arranged in an order, thus having the same number of elements (0) as N? One may think there are more rationals than positive integers, but using a very simple system, we will prove the opposite. We have to find some rule that sets up a 1-1 corres ...
... Set-up: Can the set of all rational numbers Q can be arranged in an order, thus having the same number of elements (0) as N? One may think there are more rationals than positive integers, but using a very simple system, we will prove the opposite. We have to find some rule that sets up a 1-1 corres ...
Document
... This is fundamentally different from demonstrating that if a statement S is true then a contradiction exists, which means that S must be false. In the case of Cantor's Proof we assume that the set of reals can be arranged into a list that contains all the members of the list. Then we assume that a v ...
... This is fundamentally different from demonstrating that if a statement S is true then a contradiction exists, which means that S must be false. In the case of Cantor's Proof we assume that the set of reals can be arranged into a list that contains all the members of the list. Then we assume that a v ...
Full text
... This contradicts (2) and (3). Thus, 1,37,32 are 0-linearly independent Now Schmidt's theorem shows that a2 is not algebraic. The assertion is proved. REMARK. The proposition remains true if we put xn -yn x - y where A- is a quadratic Pisot number and y its conjugate. ...
... This contradicts (2) and (3). Thus, 1,37,32 are 0-linearly independent Now Schmidt's theorem shows that a2 is not algebraic. The assertion is proved. REMARK. The proposition remains true if we put xn -yn x - y where A- is a quadratic Pisot number and y its conjugate. ...
R : M T
... Assume, for contradiction, the opposite of the statement you’re trying to prove. Then do stuff to reach a contradiction. Conclude that your assumption must be false after all. • Proof by Induction Base case: Prove the statement is true for n=1 Inductive hypothesis: Assume that the statement is true ...
... Assume, for contradiction, the opposite of the statement you’re trying to prove. Then do stuff to reach a contradiction. Conclude that your assumption must be false after all. • Proof by Induction Base case: Prove the statement is true for n=1 Inductive hypothesis: Assume that the statement is true ...
fract2
... • Cantor come to the following remarkable conclusion. He showed that one cannot “count” the reals. To see this consider how you would answer the question: “Do the real numbers form a countable set?” • What are the answers you could give? ...
... • Cantor come to the following remarkable conclusion. He showed that one cannot “count” the reals. To see this consider how you would answer the question: “Do the real numbers form a countable set?” • What are the answers you could give? ...
Section 3.3 Reading Assignment Due 9 AM, Tuesday 5/7. Please
... 3. If a first real number has decimal expansion of the form 0.2????... and a second real number has decimal expansion of the form 0.4???????..., can these two numbers be equal? Explain. ...
... 3. If a first real number has decimal expansion of the form 0.2????... and a second real number has decimal expansion of the form 0.4???????..., can these two numbers be equal? Explain. ...