Problem 23

Question

Define each recursively, where $n \geq 0$ The number $S_{n}$ of subsets of a set with $n$ elements.

Step-by-Step Solution

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Answer

The recursively defined number of subsets $S_n$ for a set with $n$ elements is given by: $S_n = \begin{cases} 1 & \text{if}\ n=0 \\ 2S_{n-1} & \text{if}\ n>0 \end{cases}$

1Step 1: Base Case

Since the base case refers to the situation when the set has no elements (empty set), we find the number of subsets when $n = 0$. There is only one subset of an empty set, which is the empty set itself. So, $S_0 = 1$.

2Step 2: Recursive Relation

Now, let's establish the recursive relation for $S_n$. Whenever a new element is added to the set, you can either add this element to the existing subsets or not. Suppose we have a set of $(n-1)$ elements with $S_{n-1}$ subsets. When we add a new element to the set, we can double the number of subsets, as each existing subset can either include the new element or not. Therefore, the relation for $S_n$ is: $S_n = 2S_{n-1}$ Putting it all together:

3Step 3: Recursive Definition of $S_n$

$S_n = \begin{cases} 1 & \text{if}\ n=0 \\ 2S_{n-1} & \text{if}\ n>0 \end{cases}$ This definition helps us determine the number of subsets for a set with $n$ elements, given the number of subsets for the previous number of elements.

Key Concepts

Base CaseRecursive RelationSubset Definition

Base Case

In the realm of recursion, the base case is the simplest possible instance of the problem that can be directly answered without further recursion. It's like the anchor that stops infinite loops! Here, the base case pertains to an empty set. The question becomes: How many subsets does an empty set have? Well, the answer is wonderfully simple. There is exactly one subset of the empty set, which is the empty set itself. This means when you have no elements, there's exactly one subset possible.
So, for a set with 0 elements (=0 ight), the number of subsets is given by $S_0 = 1$. This base case acts as the foundation upon which the entire recursive process is built. Failing to define this clear-cut starting point would leave our recursion aimless.

Recursive Relation

Understanding a recursive relation is key to unlocking the full power of recursion. This is where we define how to transition from one case to the next. Imagine you have a set with $n-1$ elements, naturally having $S_{n-1}$ subsets. But what happens when you add an additional element to create a new set with $n$ elements?
Each existing subset can welcome this new element or reject it, thereby effectively doubling the number of subsets. Therefore, if the number of subsets for $n-1$ elements is $S_{n-1}$, then for $n$ elements it's $S_n = 2S_{n-1}$.
Thanks to this clear relation, we have a powerful expression that links the number of subsets for any number of elements to the previously known subsets. This transformative step is what allows recursion to build upwards from the base case.

Subset Definition

The subset definition concept helps to envision and calculate the potential groupings within a set. A subset is any group of elements that can be derived from a larger set. When considering subsets, each element of a set can either be included in a subset or excluded.
So, if you have a set with $n$ elements, each element presents two choices: include it or not. Thus, you have $2^n$ total possible subsets. This expansive ability to consider each element contributes to the vast number of subsets possible even within relatively small sets.

An empty set has $2^0 = 1$ subset—the empty set itself.
A single-element set has $2^1 = 2$ subsets—the empty set and the set with one element.
Two elements offer $2^2 = 4$ subsets: the empty set, two individual elements, and one containing both elements.

By gradually building this understanding, the formation of the recursive relation becomes clearer and easier to apply.

Problem 23

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