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Magazine Chapter 6: Problem 6
Let \(C=\\{1,2,3,4,6,8,12,24\\}\) and define \(t\) on \(C\) by atb if and only if \(a\) and \(b\) share a common divisor greater than 1. Draw a digraph for \(t\).
Short Answer
Expert verified
Draw nodes for each element in \( C \) and connect nodes if they share a common divisor greater than 1.
Step by step solution
01
Identify Elements of C
The set \( C \) is defined as \( C=\{1,2,3,4,6,8,12,24\} \). These are the elements we have to consider for the relation \( t \).
02
Define Relation Condition
The relation \( t \) on set \( C \) is defined such that \( a \) is related to \( b \) (i.e., \( a t b \)) if and only if \( a \) and \( b \) share a common divisor greater than 1.
03
Analyze Each Pair
Consider each pair of elements from \( C \). We need to check for each pair \( (a, b) \), whether they have a common divisor greater than 1. For instance, check \( a = 2 \) and \( b = 4 \) (common divisor is 2), so \( 2t4 \).
04
List Related Pairs
After checking all pairs, the ones that meet the condition of having a common divisor greater than 1 are: \((2,4), (2,6), (2,8), (2,12), (2,24), (3,6), (3,12), (3,24), (4,6), (4,8), (4,12), (4,24), (6,8), (6,12), (6,24), (8,12), (8,24), (12,24)\) and the corresponding symmetric pairs.
05
Draw the Directed Graph
To draw the digraph, represent each element of \( C \) as a node. Draw a directed edge from node \( a \) to node \( b \) if \( (a, b) \) is a related pair. Make sure to include all pairs as listed in Step 4. Since they are undirected pairs in terms of condition, draw bidirectional edges between nodes.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Directed Graph
A directed graph, often called a digraph, is a set of nodes connected by edges, where the edges have a direction. This means each edge goes from one node to another specific node, indicating a relationship or connection.
A practical example of a directed graph is a one-way street map, where streets have direction and connect one point to another.
A practical example of a directed graph is a one-way street map, where streets have direction and connect one point to another.
- **Nodes**: In our exercise, the nodes are the elements of set \( C \), i.e., \( \{1, 2, 3, 4, 6, 8, 12, 24\} \).
- **Edges**: The edges connect nodes based on the relationship defined by a common divisor greater than 1. These edges are bidirectional because the relationship is symmetric.
Relations
In mathematics, a relation is basically a rule that connects two or more objects together. It tells us how certain elements relate to one another in a structured way.
In this exercise, we鈥檝e defined a specific relation \( t \) on the set \( C \). It states that an element \( a \) is related to \( b \) if they share a common divisor greater than 1. For example, \( 2 \) is related to \( 4 \) because they share \( 2 \) as a divisor, which is greater than 1.
In this exercise, we鈥檝e defined a specific relation \( t \) on the set \( C \). It states that an element \( a \) is related to \( b \) if they share a common divisor greater than 1. For example, \( 2 \) is related to \( 4 \) because they share \( 2 \) as a divisor, which is greater than 1.
- **Reflection**: Some relations like equality are reflexive, but in our example, not every element relates to itself unless it shares a divisor with itself like \( 8 \) does with \( 4 \).
- **Symmetry**: Our relation is symmetric, meaning if \( a \) is related to \( b \), then \( b \) is also related to \( a \).
- **Transitivity**: This would imply if \( a \) relates to \( b \) and \( b \) relates to \( c \), then \( a \) should relate to \( c \). Not all relations sustain this property naturally.
Divisors
A divisor is a number that divides another number completely without leaving a remainder. Common divisors are numbers that can divide two or more numbers fully. For example, the common divisors of \( 8 \) and \( 12 \) include \( 1, 2, \) and \( 4 \), with \( 1 \) usually being the trivial divisor.
Looking into our exercise, we have defined a relationship based on whether numbers share a divisor greater than 1. This helps in categorizing numbers by their shared properties.
Looking into our exercise, we have defined a relationship based on whether numbers share a divisor greater than 1. This helps in categorizing numbers by their shared properties.
- **Identifying Divisors**: To find if \( a \) and \( b \) are related by \( t \), we check if they have any common divisors above 1, such as \( 2 \) or \( 3 \).
- **Importance**: Understanding divisors is crucial for solving problems in number theory and has implications for cryptography, computer algorithms, and more.
Set Theory
Set theory is a fundamental part of modern mathematics and deals with collections of objects, known as sets. In the context of our exercise, the set \( C = \{1, 2, 3, 4, 6, 8, 12, 24\} \) is a collection of numbers, each considered as an element of the set.
Within set theory:
Within set theory:
- **Elements**: These are individual objects within a set. For example, \( 1 \) or \( 12 \) are elements of the set \( C \).
- **Subsets**: A subset is any set containing elements that are all members of another set. All elements related by \( t \) form a subset of the larger set \( C \).
- **Intersection**: The intersection of two sets is where both sets contain the same elements. In our case, the common elements (divisors) could be seen where relationships overlap.
