Counting

Suppose that a license plate must contain a sequence of two-letters followed by four-digits, or three-letters followed by three-digits. How many different license plates can be made?

Combinations

In how many ways can 15 distinct books be distributed so that Carol receives 6, Doc receives 4, Ellen receives 5

How many positive integer solutions are there to the equation x + y + z = 17?

How many positive integer solutions are there to the equation x + y + z = 17?

Permutations and Combinations

In how many different sequences can we list 4 novels followed by 6 biographies if there are 8 novels and 10 biographies from which to choose?

Use logic to write the negation of a sentence.

Use logic to write the negation of the sentence: All the participants of this course are IT majors or are Math majors. (Use the basic logic laws.)

Use a truth table to show a tautology

Use truth table to show (¬ q ^ (p → q)) → ¬ p is a tautology

Set Operations : Subtraction, Cartesian Product, Direct Sum and Intersection

1. Let A = {1,3,5,6} and B = {3,5} and C = {a,b,c}. Compute (a) A - B (b) B X C (X means the Cartesian product) (c) A (+) B (+) means direct sum (d) A ∩ (B - A)

Disproving an Intersection Equality by Couterexample and Writing a Converse

Let A, B, and C be any three sets and consider the sentence: If A ∩ B = A ∩ C then B = C. (a) Prove in detail that this statement is false. (Give a counterexample.) (b) Write the converse of the above statement and show through an example that it is true (no proof is necessary).

Iterations and Recurrence Relations

Zebra mussels are fresh water mollusks that attack underwater structures. Suppose that the volume of mussels in a confined area grows at a rate of 0.2% per day (1) If there are now 10 cubic feet of mussels in a lock on the Illinois River at Peoria, Illinois, develop a recurrence relation and initial conditions that represent ...continues

Augmentation Flow

A power generator at a dam can send 300 megawatts to substation 1, can send 200 megawatts to substation 2, and can send 250 megawatts to substation 2. In addition, substation 2 can send 100 megawatts to substation 1 and 70 megawatts to substation 3. Substation 1 can send at most 280 megawatts to the distribution center, and subs ...continues