Problem 8. Let Rf be the set of all real numbers with finite complement topology. In this topology G is open if it is the empty set or R – G is finite. Find the closure of [0, 1) = {x ≤R|0 ≤ x < 1} in Rf.

Q: 5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a <…

A: Since you posted a multi sub part question but according to guidelines we will solve first three sub…

Q: 6. A relation R on a set X is called a total order if it is antisymmetric, reflexive, transi- tive,…

A:

Q: You are given two graphs G and H with V(G) = {a, b, c, d} and V(H) = {1, 2, 3}. 1. How many…

A:

Q: Dijec not need to prove you function is a bijection): (a) N and Z (b) Rand (1, ∞) (c) (0, 2) and (3,…

A:

Q: Exercise 1. Let 0 < & ≤ 1, and A := [10] E (i) Calculate the condition number of A in the 1-, 2- and…

A:

Q: Let ∅ be the empty set. Let X be the real line, the entire plane, or, in the three-dimensional case,…

A: Given:

Q: 3. Use function f(x) = *** to show that Card(R – {5}) = Card(R – {1}) step by step. (Note: Card(A)…

A:

Q: a. Are any nonzero objects invisible to this operator? If so, give an example. We say that an object…

A: (1) The null space of a matrix or a linear transformation is defined as the set of vectors that when…

Q: Exercise 4. A subset A of a topological space X is called dense if cl A = X. The set Q of rational…

A:

Q: Give either an example of a function from R to R for which f(A)=B, for A and B open sets in R (open…

A: According to the question I have to construct either f or g having above properties . So I will…

Q: example 3.19- for any set ax we may defined the function X x X -&gt; X by function (u,v) = u. In…

A: The given question is solved below.

Q: 4.) If each f: IR→IR is continous then prove that { x ER : lim fx (x) = +∞o $ is the countable…

A: Given that is continuous function for each .We need to prove that  is the countable intersection of…

Q: Problem 7. A relation is defined on the set of positive integers N by n ~ m if either n divides m or…

A:

Q: Given a finite set S, find all possible values of |S] so that the set {(a, b) | (a E S) A (b e S) ^…

A:

Q: (a) Suppose S is a subset of positive integers, ZZ. If |S| ≥ 3, prove there exist distinct z and y…

A: (a) Given that S is subset of  such that Observed that any positive integer is either even or odd…

Q: Let A = {2, 3, 4, . . .} be ordered by “x divides y”. (i.) Determine the minimal elements of A.…

A: Let  be ordered by “ divides ”.(i). Determine the minimal elements of .(ii). Determine the maximal…

Q: Given two sets X and Y, we denote the cardinality of X by |X| and the set of all functions from X to…

A: We are given two sets X and Y. The set of all functions from X to Y is denoted by YX.  Suppose that…

Q: ) A) Show that the set of all odd integers is countable, by producing a 1-1 unction F From J onto…

A:

Q: R.3. Let [a, b] be an interval on the real number line (with a < b). Prove that a,b]-[0,1].

A: We have to find a continuous bijective from a,b to 0,1 to show that a,b~0,1

Q: Suppose E is a bounded, nonvoid set of real numbers. Define |E| to be the set { x: x = E}. Is it…

A:

Q: Let A = {x | 5x + 1 E Q} and B be the set of all even integers. Prove that A and B have the same…

A:

Q: (a) Represent the set {x = (x1, x2) = R² | x1x2 ≥ 1}, as the intersection of some family of…

A: The given set is x=(x1,x2)∈R+2|x1x2≥1. To Find: a) Represent the set x=(x1,x2)∈R+2|x1x2≥1 as the…

Q: 6.3. Let (A, <A) and (B,<B) be two partially ordered sets. One can define on the cartesian product A…

A: Given: A, ⩽A and B, ⩽B are partially ordered sets.  A Cartesian product A x B is defined with…

Q: Give a binary relation R such that R is irreflexive, but R2 is not. You may chose the set over R is…

A: Irreflexive and reflexive relationship