By Alan Doerr, Kenneth Levasseur

ISBN-10: 1365355942

ISBN-13: 9781365355943

Textbook from UMass Lowell, model 3.0

Creative Commons License

Applied Discrete constructions by means of Alan Doerr & Kenneth Levasseur is approved below an artistic Commons Attribution-NonCommercial-ShareAlike 3.0 usa License.

Link to professor's web page: http://faculty.uml.edu/klevasseur/ads2/

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**Example text**

Let m be a positive integer with n-bit binary representation: an 1 an 2 · · · a1 a0 with an 1 = 1 What are the smallest and largest values that m could have? 7. If a positive integer is a multiple of 100, we can identify this fact from its decimal representation, since it will end with two zeros. What can you say about a positive integer if its binary representation ends with two zeros? What if i ends in k zeros? 8. Can a multiple of ten be easily identified from its binary representation? 5 Summation Notation and Generalizations Most operations such as addition of numbers are introduced as binary operations.

1 ( Cartesian Product). Let A and B be sets. The Cartesian product of A and B, denoted by A ⇥ B, is defined as follows: A ⇥ B = {(a, b) | a 2 A and b 2 B}, that is, A ⇥ B is the set of all possible ordered pairs whose first component comes from A and whose second component comes from B. 2 (Some Cartesian Products). Notation in mathematics is often developed for good reason. In this case, a few examples will make clear why the symbol ⇥ is used for Cartesian products. • Let A = {1, 2, 3} and B = {4, 5}.

If ten faculty members and 25 students have been nominated for the committee, how many judiciary committees could be formed at this point ? 2. Suppose that a single character is stored in a computer using eight bits. a. How many bit patterns have exactly three 1’s? b. How many bit patterns have at least two 1’s? Hint. Think of the set of positions that contain a 1 to turn this is into a question about sets. 3. How many subsets of {1, 2, 3, . . , 10} contain at least seven elements? 4. The congressional committees on mathematics and computer science are made up of five congressmen each, and a congressional rule is that the two committees must be disjoint.

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