Find (a) the number of subsets and (b) the number of proper subsets of the set. The set of days of the week. .
The Correct Answer and Explanation is:
Let’s break down the problem step by step for the set of days of the week, which is represented as:S={Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}S = \{ \text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} \}S={Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
(a) Number of subsets:
The number of subsets of a set is given by the formula:Number of subsets=2n\text{Number of subsets} = 2^nNumber of subsets=2n
where nnn is the number of elements in the set. In this case, there are 7 elements (the 7 days of the week), so n=7n = 7n=7.Number of subsets=27=128\text{Number of subsets} = 2^7 = 128Number of subsets=27=128
This means there are 128 subsets of the set of days of the week, including the empty set and the set itself.
(b) Number of proper subsets:
A proper subset is a subset that is not identical to the original set. The formula for the number of proper subsets is:Number of proper subsets=2n−1\text{Number of proper subsets} = 2^n – 1Number of proper subsets=2n−1
This is because we exclude the set itself from the total number of subsets. Using n=7n = 7n=7 again:Number of proper subsets=27−1=128−1=127\text{Number of proper subsets} = 2^7 – 1 = 128 – 1 = 127Number of proper subsets=27−1=128−1=127
Therefore, there are 127 proper subsets of the set of days of the week.
Explanation:
- Subsets: A subset of a set is any collection of elements from the set, including the empty set (which contains no elements) and the set itself. For a set with nnn elements, there are 2n2^n2n possible subsets because each element can either be included or excluded in a subset.
- Proper subsets: A proper subset is any subset that does not equal the original set. Since the set of all subsets (which includes the set itself) has 2n2^n2n subsets, we subtract one to exclude the set itself, leaving 2n−12^n – 12n−1 proper subsets.
So, the answers are:
- (a) The number of subsets is 128.
- (b) The number of proper subsets is 127.
