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Chapter 3: Problem 14

a) How many subsets of \(\\{1,2,3, \ldots, 11\\}\) contain at least one even integer? b) How many subsets of \(\\{1,2,3, \ldots, 12\\}\) contain at least one even integer? c) Generalize the results of parts (a) and (b).

Short Answer

Expert verified
a) The total subsets containing at least one even integer for the set {1, 2, 3, ..., 11} is \(2^{11} - 2^6 = 2016\). b) For the set {1, 2, 3, ..., 12}, the total subsets is \(2^{12} - 2^6 = 4032\). c) Generalized formula: For a set {1, 2, 3, ...., n} with n>=2 the number of subsets with at least one even number is \(2^n - 2^{\frac{n}{2}}\) for even n and \(2^n - 2^{\frac{n-1}{2}}\) for odd n.

Step by step solution

01

Identify Total Subsets

First, calculate the total number of subsets for the set {1, 2, 3, ...., 11}. This is found by \(2^n\), where n is the number of elements in the set. So for 11 elements, we get \(2^{11}\) total subsets.
02

Find Subsets Without Any Even Numbers

Now, find the total number of subsets that do not contain an even number. This would be subsets of odd numbers only, {1, 3, 5, 7, 9, 11}, a set of 6 elements. Using the same formula, we get \(2^6\) subsets.
03

Subtract To Find Subsets with Even Numbers

To find the total number of subsets that contain at least one even number, subtract the total subsets of odd numbers from the total subsets. Thus the result will be \(2^{11} - 2^6\). Repeat the same process for the set {1, 2, 3, ...., 12}.
04

Generalize the Results

On the basis of observations in step 1 to 3 for the cases 11 and 12, we can generalize that the number of subsets with at least one even number for a set {1, 2, 3, ...., n} with n>=2 is given by the formula \(2^n - 2^{\frac{n}{2}}\) for even n and \(2^n - 2^{\frac{n-1}{2}}\) for odd n.

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