Let \(A=\\{\mathrm{H}, \mathrm{T}\\}, B=\\{1,2,3,4,5,6\\},\) and \(C=\\{\) red, green blue \(\\}\). Find the numbers indicated. \(n(B \times B)\)

A binary digit, or "bit," is either 0 or \(1 .\) A nybble is a four-bit sequence. How many different nybbles containing a single 1 are possible?

Use Venn diagrams to illustrate the given identity for subsets \(A, B,\) and \(C\) of \(S\). \((A \cup B) \cup C=A \cup(B \cup C) \quad\) Associative law

Theory of Linear Programming (Some familiarity with linear programming is assumed for this exercise.) Suppose you have a linear programming problem with two unknowns and 20 constraints. You decide that graphing the feasible region would take a lot of work, but then you recall that corner points are obtained by solving a system of two equations in two unknowns obtained from two of the constraints. Thus, you decide that it might pay instead to locate all the possible corner points by solving all possible combinations of two equations and then checking whether each solution is a feasible point. a. How many systems of two equations in two unknowns will you be required to solve? b. Generalize this to \(n\) constraints.

Let \(S\) be the set of outcomes when two distinguishable dice are rolled, let \(E\) be the subset of outcomes in which at least one die shows an even number, and let \(F\) be the subset of outcomes in which at least one die shows an odd number. List the elements in the given subset. \(E^{\prime} \cap F^{\prime}\)

Databases A freelance computer consultant keeps a database of her clients, which contains the names \(S=\\{\) Acme, Brothers, Crafts, Dion, Effigy, Floyd, Global, Hilbert \(\\}\) The following clients owe her money: \(A=\\{\) Acme, Crafts, Effigy, Global \(\\}\) The following clients have done at least \(\$ 10,000\) worth of business with her: \(B=\\{\) Acme, Brothers, Crafts, Dion \(\\} .\) The following clients have employed her in the last year: \(C=\\{\) Acme, Crafts, Dion, Effigy, Global, Hilbert \(\\}\). A subset of clients is described that the consultant could find using her database. Write the subset in terms of \(A, B,\) and \(C,\) and list the clients in that subset. [HINT: See Example \(4.]\) The clients who do not owe her money or have employed her in the last year