Suppose that A is a set and {B_i| i I} is an indexed families of sets. Prove that

A(U_iIB_i) = U_iI(AB_i).

Suppose that A = {1, 2, 3}, B = {4, 5}, C = {a, b, c, d}, R = {(1, b), (2, a), (2, b), (2, c), (3, d)} and S = {(4, a), (4, d), (5, b), (5, c)}. Note that R is a relation from A to C and S is a relation from B to C.

a.Find S^-1R. This is a relation from which set to which other set? Justify your solution.

b.Find R^-1S. This is a relation from which set to which other set? Justify your solution.

Suppose R and S are relations from A to B. Must the following statements be true? Justify your answers with proofs or counterexamples.

a.R = Dom(R) Ran(R)

b.(R S)^-1= R^-1 S^-1

List the ordered pairs in the relations represented by the following graph. Determine whether this relation is reflexive, symmetric, or transitive.Justify your answers with reasoning or counterexamples.

Consider the function f: defined on positive integers with f(n) = n "flipped" as a mirror image into a decimal. For example, f(5) = .5, f(418) = .814, and f(1000) = .0001. Define a relation R on the positive integers as (m, n) R if and only if f(m) f(n). For example, (5, 418) R because .5 .814 but (418, .923) R because .814 > .329. Is R a partial order?Either provide a proof to show that this is true or provide a counterexample to show that this is false.

Define a relation R on as (a, b) R if and only if a and b, when written out, have the same number of 5s. For example, (1752, 95) R since they both have one 5 but (1752, 505) R since 1752 has one 5 but 505 has two 5s. Is R an equivalence relation?Prove that R is an equivalence relation.