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1solution - ECEN 303 Assignment 1 Problems 1 Following the...

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ECEN 303: Assignment 1 Problems: 1. Following the argument presented in the notes, prove that parenleftBigg intersectiondisplay α A S α parenrightBigg c = uniondisplay α A S c α . Suppose that x belongs to (intersectiontext α I S α ) c . That is, there exists an α I such that x is not an element of S α . This implies that x belongs to S c α , and therefore x uniontext α I S c α . Thus, we have shown that (intersectiontext α I S α ) c uniontext α I S c α . The converse is obtained by reversing the argument. Suppose that x belongs to uniontext α I S c α . Then, there exists an α I such that x S c α . This implies that x / S α and, as such, x / (intersectiontext α I S α ) . Alternatively, we have x (intersectiontext α I S α ) 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? How
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