Recitation 13 Sum and product rules Problem 1. A license plate consists of either: 3 letters followed by 3 digits(standard plate 5 letters(vanity plate) Let L be the set of all possible license plates (a)Express L in terms of A={A,B,C,…,公} {0,1,2 using unions (U) and set products(x) Solution L=(A3×D3)∪A5 rules. pute Ll, the number of different license plates, using the sum and product Solution L|=|(A3×D3)UA5 (A3×D>)+|A° Sum rule A.D+Al Product rule =263·103+265 Bijections Problem 2. For each part below, describe a bijection between the two sets mentioned. The existence of such a bijection proves that the two sets are the same size. a good approach is to describe an element of the first set using variables and then describe the corresponding element of the second set in terms of those variables. For example, we might describe a bijecton from ways of selecting a dozen doughnuts from five varieties to a 16-bit string with four 1s as follows:� � � � � � � � � � � � Recitation 13 2 Sum and Product Rules Problem 1. A license plate consists of either: • 3 letters followed by 3 digits (standard plate) • 5 letters (vanity plate) Let L be the set of all possible license plates. (a) Express L in terms of A = {A, B, C, . . . , Z} D = {0, 1, 2, . . . , 9} using unions ( ) ∪ and set products (×). Solution. L = (A3 × D3 ) ∪ A5 (b) Compute |L|, the number of different license plates, using the sum and product rules. Solution. (A3 × D3 ) ∪ A5 A5 | | L = = (A Sum Rule 3 × D3 ) + 3 3 5 = |A| · |D| | + |A Product Rule = 263 103 + 265 · Bijections Problem 2. For each part below, describe a bijection between the two sets mentioned. The existence of such a bijection proves that the two sets are the same size. A good approach is to describe an element of the first set using variables and then describe the corresponding element of the second set in terms of those variables. For example, we might describe a bijecton from ways of selecting a dozen doughnuts from five varieties to a 16­bit string with four 1’s as follows: