Let X and Y be non-empty sets. If A1 and A2 are subsets of X, and B1 and B2 subsets of Y. Show that (A1×B1)U(A2×B2) = (A1UA2)×(B1UB2).

Topology Sets and Functions (XLIV) Functions Let X and Y be non-empty sets. If A1 and A2 are subsets of X and B1and B2 subsets of Y, show that (A1×B1)U ...continues

Let f:X→Y be an arbitrary mapping. Define a relation in X as follows: x1 ~ x2 means that f(x1) = f(x2). Show that this is an equivalence relation and describe the equivalence sets.

Topology Sets and Functions (XLV) Functions Let f:X→Y be an arbitrary mapping. Define a relation in X as follows: x1 ~ x2 means that f(x1) = f(x2). Show tha ...continues

In the set R of all real numbers, let x ~ y means that x – y is an integer. Show that this is an equivalence relation and describe the equivalence sets.

Topology Sets and Functions (XLVI) Functions In the set R of all real numbers, let x ~ y means that x – y is an integer. Show that this i ...continues

Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by a ≡ b (mod m) - if a – b is exactly divisible by m, i.e., if a – b is an integral multiple of m. Show that this is an equivalence relation , describe the equivalence set, and state the number of distinct equivalence sets.

Topology Sets and Functions (XLVII) Functions Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by a ...continues

Decide which ones of the three properties of reflexivity, symmetry, and transitivity are true for each of the following relations in the set of all positive integers: m ≤ n, m < n, m divides n. Are any of these equivalence relations?

Topology Sets and Functions (XLVIII) Functions Decide which ones of the three properties of reflexivity, symmetry, and transitivity are true for each of the following relations in ...continues

Difference Between Measurable Functions

Please see the attached file for the fully formatted problems. Given: and and is measurable and is a null set. 1) is zero except on the null set, true of false? 2) where is a null set, true or false? I have been told that both 1 and 2 are true, but I don't understand why. Since the set {x: f(x)=g(x)} is null, ...continues

Determine whether the set of cylinders of the given pair of measure spaces is contained in (is a subset of) the set of rectangles of that pair of measure spaces.

Let (Omega_1, F_1, P_1) and (Omega_2, F_2, P_2) be the following measure spaces: Omega_1 = {a, b}, F_1 is the sigma algebra of all subsets of Omega_1, and P_1 is a measure on Omega_1. Omega_2 = {c, d}, F_2 is the sigma algebra of all subsets of Omega_2, and P_2 is a measure on Omega_2. Determine the makeup of the set C ...continues

Convergence and Uniform Convergence

see attached. Let f_n(x):R-->Rbe the function... Show that f_n(x)-->f(x) for each x in R Show that f_n does not converge uniformly to f.

Retractions

Let A_0 be contained in A_1 contained in A_2 and so on be a nested sequence of subspaces of X such that the union of all A_n is X and such that An contained in the interior of A_(n+1). Suppose for each n, there is a retraction r_n:A_(n+1) to An. Prove there is a retraction r: X to A_0.

Inclusion map

Show that the inclusion map i:Q -> R defined by i(q)=q for all q in Q, is continuous where both Q (rational numbers) and R(real numbers) are given the order topology.