By Alan Doerr, Kenneth Levasseur

Textbook from UMass Lowell, model 3.0

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Applied Discrete constructions through Alan Doerr & Kenneth Levasseur is authorized lower than 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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**Extra info for Applied Discrete Structures for Computer Science**

**Example text**

The ease with which we can prove this fact demonstrates the power and usefulness of the rule of products. Do not underestimate the usefulness of simple ideas. 7 (Power Set Cardinality Theorem). If A is a finite set, then |P(A)| = 2|A| . 26 CHAPTER 2. COMBINATORICS Proof. Proof: Consider how we might determine any B 2 P(A), where |A| = n. For each element x 2 A there are two choices, either x 2 B or x 2 / B. Since there are n elements of A we have, by the rule of products, 2 · 2 · · · · · 2 = 2n n factors diﬀerent subsets of A.

How many diﬀerent bit patterns are there? ) (b) How many bit patterns are palindromes (the same backwards as forwards)? (c) How many diﬀerent bit patterns have an even number of 1’s? 1. BASIC COUNTING TECHNIQUES - THE RULE OF PRODUCTS 27 10. Automobile license plates in Massachusetts usually consist of three digits followed by three letters. The first digit is never zero. How many diﬀerent plates of this type could be made? 11. (a) Let A = {1, 2, 3, 4}. Determine the number of diﬀerent subsets of A.

PARTITIONS OF SETS AND THE LAW OF ADDITION 35 2. |A1 [ A2 [ A3 | = |A1 | + |A2 | + |A3 | (|A1 \ A2 | + |A1 \ A3 | + |A2 \ A3 |) + |A1 \ A2 \ A3 | The inclusion-exclusion laws extend to more than three sets, as will be explored in the exercises. In this section we saw that being able to partition a set into disjoint subsets gives rise to a handy counting technique. Given a set, there are many ways to partition depending on what one would wish to accomplish. One natural partitioning of sets is apparent when one draws a Venn diagram.