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Chapter 8: Problem 23

a) Find the recurrence relation satisfied by \(S_{n},\) where \(S_{n}\) is the number of regions into which three-dimensional space is divided by \(n\) planes if every three of the planes meet in one point, but no four of the planes go through the same point, b) Find \(S_{n}\) using iteration.

Short Answer

Expert verified
The recurrence relation is \(S_n = S_{n-1} + \frac{n(n+1)}{2}\) and iterating it gives, \(S_0 = 1\), \(S_1 = 2\), \(S_2 = 4\), \(S_3 = 8\).

Step by step solution

01

- Understanding the problem

To find the recurrence relation, identify how adding each new plane affects the number of regions in three-dimensional space.
02

- Initial conditions

For no planes (n=0), the number of regions is clearly 1: \[S_0 = 1\]. For one plane (n=1), it divides the space into two regions: \[S_1 = 2\].
03

- Adding a second plane

With two planes (n=2), if they intersect, they divide the space into four regions: \[S_2 = 4\].
04

- Adding a third plane

For three planes (n=3), each of which intersects the other two, the three planes intersect in such a way as to divide space into 8 regions: \[S_3 = 8\].
05

- General recurrence relation

Notice the pattern when adding a new plane: Each new plane intersects all previous planes, resulting in regions being added. Each new plane adds regions based on previous intersections: \[S_n = S_{n-1} + n(n-1)/2 + n + 1\].
06

- Simplifying the recurrence relation

Combine terms to simplify the recurrence relation: \[S_n = S_{n-1} + \frac{n(n+1)}{2} \].
07

- Using iteration

Start from the initial condition and iterate using the recurrence relation. Using the recurrence, from \(S_0 = 1\), we can compute: \[S_1 = 1 + \frac{1(2)}{2} = 2\]\[S_2 = 2 + \frac{2(3)}{2} = 4\]\[S_3 = 4 + \frac{3(4)}{2} = 8\]Continue iterating to find higher values of \(S_n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Mathematics
Discrete mathematics focuses on structures that are countable or distinct and separate. It covers a variety of topics, including logic, set theory, graph theory, and recurrence relations. Recurrence relations are equations that define sequences recursively: each term is a function of the preceding terms. In our exercise, we are interested in finding how many regions in a three-dimensional space are created by intersecting planes. We express this number using a recurrence relation. A good understanding of discrete mathematics is essential for formulating and solving recurrence relations, as well as for recognizing patterns in sequences and structures.

To dive deeper, consider that when we add planes one by one, the number of additional regions created depends on the previous number of planes and their intersections. This kind of problem exemplifies the beauty and usefulness of discrete mathematics in solving real-world problems by breaking them down into manageable, countable parts.
Iterative Methods
Iterative methods involve solving a problem through repeated application of a formula or computational process. In the context of our problem, iteration is used to find higher terms of the sequence defined by our recurrence relation. Starting from an initial condition, we can use the recurrence relation to compute successive terms. Here鈥檚 how we applied iterative methods:

1. Start with known values:
  • For no planes (=0), there is 1 region: S_0<=1
  • For one plane (=1), the space is divided into 2 regions: S 1<=2
  • For two planes (_2\text{) }), when they intersect, the space is divided into 4 regions: S__2=4.

    • 3. Using recurrence formula \(\ S_n = S_n_1
    \lfrac*n.1-n) {s+2})\) computeNext ter S. The process continues but noting each ter refers ones before it Until given numberFor students iterative knowing importance recursion in computer algorithms divide-conquer strategy is critical Messages decomposition-systematic-solvingchunk problems.
Three-Dimensional Space
Understanding three-dimensional space is crucial for visualizing how planes divide space into regions. The concept involves examining how objects exist and interact within a volume, defined by three axes: x, y, and z. Here, each new plane added intersects all previous planes, forming a complex network of regions.

Here are key aspects of three-dimensional space related to our problem:
  • **Points**: Locations in space defined by coordinates (x, y, z).
  • **Planes**: Flat surfaces extending infinitely in space, described by equations like ax + by + cz = d.
  • **Intersections**: When planes intersect, they form lines, and when multiple planes intersect, they create more confined regions.
In our problem, each new plane intersects with all previously added planes, further subdividing the existing regions into more regions. Understanding the spatial relationships and how intersections work in three dimensions helps in formulating and visualizing the recurrence relation for the number of regions formed. For example, with three planes, each plane intersects the other two, creating multiple new regions within the three-dimensional space. Visualizing these intersections can greatly aid in understanding the patterns formed and the resulting recursive formula.

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