Applied Discrete Structures for Computer Science by Alan Doerr, Kenneth Levasseur

By Alan Doerr, Kenneth Levasseur

Textbook from UMass Lowell, model 3.0

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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.

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