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Chapter 2: Problem 13

Let \(X=\\{0,1,2, \ldots, 16\\}\). Express each of the following permutations of \(X\) as a product of disjoint cycles: (a) the function \(f_{1}\) defined by \(f_{1}(x) \equiv x+5\) mod 17; (b) the function \(f_{2}\) defined by \(f_{2}(x) \equiv 2 x \bmod 17\); (c) the function \(f_{3}\) defined by \(f_{3}(x) \equiv 3 x+1\) mod 17 .

Short Answer

Expert verified
(a) A single cycle: (0 5 10 15 3 8 13 1 6 11 16 4 9 14 2 7 12), (b) (0) (1 2 4 8 16 15 13 9) (3 6 12 7 14 11 5 10), (c) (0 1 4 13 6 2 7 5 16 15 11) (3 10 12) (8 9).

Step by step solution

01

Understanding disjoint cycles

A disjoint cycle is a permutation of a finite set that can be broken down into independent cycles where each cycle does not share any elements with another cycle. Each cycle represents a sequence or loop of numbers where applying the permutation returns each number in the cycle to another number in the cycle, eventually cycling back to the start.
02

Expressing \(f_1\) as a Cycle

The function \(f_1(x) \equiv x+5 \mod 17\) is a permutation of set \(X\). Begin with an element, say 0, and repeatedly apply \(f_1\) until you return to the starting element. The numbers visited form a cycle: \(0 \to 5 \to 10 \to 15 \to 3 \to 8 \to 13 \to 1 \to 6 \to 11 \to 16 \to 4 \to 9 \to 14 \to 2 \to 7 \to 12 \to 0\). Since all elements of \(X\) are used, this is a single 17-cycle.
03

Expressing \(f_2\) as a Cycle

The function \(f_2(x) \equiv 2x \mod 17\) is applied to each element of \(X\). Start from 0, getting the cycle \(0\). Next, starting from 1: \(1 \to 2 \to 4 \to 8 \to 16 \to 15 \to 13 \to 9 \to 1\), forming a cycle. Starting from 3: \(3 \to 6 \to 12 \to 7 \to 14 \to 11 \to 5 \to 10 \to 3\), forming another cycle. So, the disjoint cycles are \((0), (1 \ 2 \ 4 \ 8 \ 16 \ 15 \ 13 \ 9), (3 \ 6 \ 12 \ 7 \ 14 \ 11 \ 5 \ 10)\).
04

Expressing \(f_3\) as a Cycle

Using \(f_3(x) \equiv 3x + 1 \mod 17\), start with 0, yielding the cycle \(0 \to 1 \to 4 \to 13 \to 6 \to 2 \to 7 \to 5 \to 16 \to 15 \to 11 \to 0\). The other numbers are cycled as \(3 \to 10 \to 12 \to 3\), forming \( (3 \ 10 \ 12) \), and \(8 \to 9 \to 8\), forming \( (8 \ 9) \). The full disjoint cycle representation includes \( (0 \ 1 \ 4 \ 13 \ 6 \ 2 \ 7 \ 5 \ 16 \ 15 \ 11) \), \((3 \ 10 \ 12)\), and \((8 \ 9)\).

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Key Concepts

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

Disjoint Cycles
Disjoint cycles are a neat way to express permutations by breaking them down into independent groups. Each cycle only cycles through unique elements of a set, without overlap. Imagine you are organizing a round-robin tournament where each player plays with a distinct set of others, cycling back to the start once everyone has played. This is akin to how disjoint cycles function.
  • A permutation can have one or more disjoint cycles.
  • Disjoint cycles offer a clear view of how elements reposition.
  • Cycles can be drawn as sequences, looping back to the initial element.
For example, with permutation function \(f_1(x) = x + 5\; \text{mod}\; 17\), starting from 0, each element maps onto the next. Here, every element from 0 to 16 is included in the single cycle, simplifying notation and giving us a comprehensive view of the movement in the cycle.
Modular Arithmetic
Modular arithmetic is a type of mathematics that deals with integers where numbers wrap around upon reaching a certain value, known as the modulus. Consider it as the number wheel of a clock where, after hitting 12, it starts back at 1.
  • This wrapping effect is essential for working with computations in a cyclic manner.
  • In the exercise, modulus 17 ensures calculations circle back to 0 after reaching 16.
  • It鈥檚 particularly prevalent in computing, cryptography, and when dealing with finite sets.
For instance, the permutation function \(f_2(x) = 2x \;\text{mod}\; 17\) uses this principle to map elements, ensuring that arithmetic remains within the set \(\{0, 1, 2, \ldots, 16\}\). This method preserves structure and order within cycles.
Finite Sets
Finite sets are collections of distinct elements that have a specific number, or cardinality, of items. They are the backbone in many fields like set theory, probability, and even computer science.
  • These sets have a clear beginning and end, making them predictable and manageable.
  • In our context, a set like \(X = \{0, 1, 2, \ldots, 16\}\) is a finite set with 17 elements.
  • Finite sets provide a controlled environment to apply permutations, ensuring comprehensive coverage of all elements.
Understanding finite sets aids in comprehending the full behavior of permutations like \(f_3(x) = 3x + 1 \;\text{mod}\; 17\), where each element undergoes exactly one transformation, eventually ensuring all are returned back to their respective cycle.

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