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Unformatted text preview: ECEN 303: Assignment 1 Problems: 1. Following the argument presented in the notes, prove that parenleftBigg intersectiondisplay A S parenrightBigg c = uniondisplay A S c . Suppose that x belongs to (intersectiontext I S ) c . That is, there exists an I such that x is not an element of S . This implies that x belongs to S c , and therefore x uniontext I S c . Thus, we have shown that (intersectiontext I S ) c uniontext I S c . The converse is obtained by reversing the argument. Suppose that x belongs to uniontext I S c . Then, there exists an I such that x S c . This implies that x / S and, as such, x / (intersectiontext I S ) . Alternatively, we have x (intersectiontext I S ) c . Putting these two inclusions together, we get the desired result. 2. John, Jim, Jay, and Jack have formed a band consisting of 4 instruments. If each of the boys can play all 4 instruments, how many different arrangements are possible? What if John and Jim can play all 4 instruments, but Jay and Jack can each play only piano and drums? There are 4! different arrangements when each of the boys can play all 4 instruments. On there other hand, there are only 4 possible arrangements in the second scenario. 3. For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9; the second digit was either 0 or 1; the third digit was any integer between 1 and 9. How many area codes were possible? How1; the third digit was any integer between 1 and 9....
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This note was uploaded on 02/11/2010 for the course ECEN 303 taught by Professor Chamberlain during the Fall '07 term at Texas A&M.
- Fall '07