hw2-solns - Illinois Institute of Technology Department of...

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Illinois Institute of Technology Department of Computer Science Solutions to Homework Assignment 2 CS 330 Discrete Structures Spring Semester, 2008 1. Page 346, Problem 58. Use the product rule to show that there are 2 2 n different truth tables for propositions in n variables. There are 2 n rows in a truth table since each of the n variables can take on two values ( TorF ), i.e., cartesian product. Each row in the table can take on two values, so there are a total of 2 2 n possible truth tables. For example, let the following table represent represent the 16 possible truth tables γ 1 . . . γ 16 for a proposition with n = 2 variables a and b . a b γ 1 γ 2 γ ... γ 16 F F F T . . . T F T F F . . . T T F F F . . . T T T F F . . . T 2. Page 362, Problem 30. Seven women and nine men are on the faculty in the math department. (a) How many ways are there to select a committee of five members of the department if at least one woman must be on the committee? There are ( 7+9 5 ) = ( 16 5 ) ways to select a committee if there are no restrictions about gender. There are ( 9 5 ) ways to select a committee of just 5 men. Therefore by removing the invalid set (all men), from the unrestricted set (any combination of men or women) there are ( 16 5 ) - ( 9 5 ) = 4638 - 126 = 4242 ways to select a committee with at least one woman.
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