4. Show that a set A is infinite if and only if it is equivalent to...
7. (5 points) Show that the following set is infinite by placing into a one-to-one correspondence with a proper subset of itself. Show the general terms of the set and subset. (a) [3,6,9,12,15...
1. Prove that any infinite set contains a countable subset (see Problem 20, page 43)
Prove that a subset of a countably infinite set is finite or countably infinite.
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection between A and some infinite subset of N.)
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection...
6.1.3. Consider the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. (Such a matrix has entries aij, where i and j are natural numbers. For each such matrix, there is a natural number n such that aij 0 ifi-n or j 〉 n.) Show that the set of such matrices is a ring without identity element.
6.1.3. Consider the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. (Such...
Let X be a non-empty set. Show that the only dense subset of X with respect to the discrete metric ddise is X. The whole set of any metric spaces is always dense, so this question is really asking you to exclude all other possibilities. Show that if (X, d) is a metric space and has dense subset A + X, then (X, d) is not topologically equivalent to (X, ddisc). (Note that this is another way of showing that...
8. Show: An infinite set of points need not be a closed set.
6. Let X be an infinite set and let U = {0}U{A CX :X \ A is finite}. (a) Prove that U is a topology on X. (b) Let B be an infinite subset of X. What is the set of limit points (also known as "accumulation points") of B? (c) Let B be a finite subset of X. What is the set of limit points of B?
Suppose |N| ≤ |S|, or in other words, S contains a countably infinite subset. Show that there exists a countably infinite subset A ⊂ S and a bijection between S \A and S.
3. Suppose S is a linearly independent generating set for a vector space V . Show that S is an efficient generating set, i.e., any proper subset of S is not a generating set.