Let S = {2, 4, 6, 8, ______ 20}.
1024
Understanding the Set S and Finding its Elements
The given set is S = {2, 4, 6, 8, ______, 20}. This set contains even numbers starting from 2 and going up to 20. To find the maximum number of subsets this set can have, we first need to determine the number of elements in the set S.
Let's list the elements of the set S explicitly:
- The first element is 2 (\(2 \times 1\)).
- The second element is 4 (\(2 \times 2\)).
- The third element is 6 (\(2 \times 3\)).
- ...
- The last element is 20 (\(2 \times 10\)).
We can see that each element in the set S is of the form \(2 \times k\), where k is a positive integer starting from 1. The elements are \(2 \times 1, 2 \times 2, 2 \times 3, \dots, 2 \times 10\). This means the values of k range from 1 to 10.
Therefore, the set S can be written as S = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}.
By counting the elements, we find that the set S has 10 elements.
Let 'n' be the number of elements in set S. In this case, \(n = 10\).
Calculating the Maximum Number of Subsets
The maximum number of subsets that a set can have is determined by the number of elements it contains. If a set has 'n' elements, the total number of subsets it can have is given by the formula \(2^n\). This includes the empty set and the set itself.
For our set S, which has \(n = 10\) elements, the total number of subsets is \(2^{10}\).
Let's calculate \(2^{10}\):
\(2^{1} = 2\)
\(2^{2} = 2 \times 2 = 4\)
\(2^{3} = 4 \times 2 = 8\)
\(2^{4} = 8 \times 2 = 16\)
\(2^{5} = 16 \times 2 = 32\)
\(2^{6} = 32 \times 2 = 64\)
\(2^{7} = 64 \times 2 = 128\)
\(2^{8} = 128 \times 2 = 256\)
\(2^{9} = 256 \times 2 = 512\)
\(2^{10} = 512 \times 2 = 1024\)
So, the maximum number of subsets that the set S has is 1024.
Understanding the Concept of Subsets
A subset is a set containing some or all of the elements of another set. For example, if A = {1, 2}, its subsets are {}, {1}, {2}, and {1, 2}. There are \(2^2 = 4\) subsets. This simple example illustrates the formula \(2^n\) in action.
Revision Table: Subsets of Sets
| Set | Elements | Number of Elements (n) | Number of Subsets (\(2^n\)) |
|---|---|---|---|
| {} (Empty Set) | None | 0 | \(2^0 = 1\) |
| {a} | a | 1 | \(2^1 = 2\) |
| {a, b} | a, b | 2 | \(2^2 = 4\) |
| {a, b, c} | a, b, c | 3 | \(2^3 = 8\) |
| S = {2, 4, ..., 20} | 2, 4, ..., 20 | 10 | \(2^{10} = 1024\) |
Additional Information: Power Sets
The collection of all possible subsets of a given set S is called the power set of S. It is often denoted by P(S) or \(2^S\). The number of elements in the power set P(S) is equal to the total number of subsets of S, which is \(2^n\), where n is the number of elements in S.
For our set S = {2, 4, ..., 20}, the power set P(S) contains 1024 subsets.
We can also talk about proper subsets. A proper subset of S is any subset of S except S itself. The number of proper subsets of a set with n elements is \(2^n - 1\). In the case of set S, the number of proper subsets would be 1024 - 1 = 1023. However, the question asks for the maximum number of subsets, which includes the set S itself.
Therefore, the maximum number of subsets for the set S is 1024.
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