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Chapter 6: Problem 335

Draw the digraph of the relation on the set \\{Sam, Mary, Pat, Ann, Polly, Sarah\\} given by "has the same first letter as."

Short Answer

Expert verified
Vertices are Sam, Mary, Pat, Ann, Polly, Sarah. Edges are: Sam 鈫� Sarah, Pat 鈫� Polly.

Step by step solution

01

- Understand the relation

The relation 'has the same first letter as' means that there will be a directed edge from one person to another if both names start with the same letter.
02

- Identify the first letters of the names

List the first letters of each name:- Sam (S)- Mary (M)- Pat (P)- Ann (A)- Polly (P)- Sarah (S)
03

- Group names with the same first letter

Group the names based on their first letters:- A: Ann- M: Mary- P: Pat, Polly- S: Sam, Sarah
04

- Draw vertices for each name

Draw a vertex for each name: Sam, Mary, Pat, Ann, Polly, Sarah.
05

- Draw directed edges based on the relation

Draw directed edges between the names that share the same first letter:- From Sam to Sarah- From Sarah to Sam- From Pat to Polly- From Polly to Pat
06

- Verify the digraph

Ensure all names with the same first letter have directed edges to and from each other. The final digraph should show vertices: Sam to Sarah (and converse), Pat to Polly (and converse). Ann and Mary stand alone as they do not share their first letter with any other names.

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Key Concepts

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

digraph
A digraph, or directed graph, is a type of graph where the edges have a direction. This means that each edge goes from one vertex to another and has an orientation.
In the context of the given exercise, a digraph is used to represent a relation where an arrow points from one name to another. This arrow indicates that both names start with the same letter.

When constructing the digraph, each person's name is a vertex. An arrow (directed edge) connects these vertices if the first letter of both names is the same. For example, since 'Sam' and 'Sarah' start with 'S', there will be arrows from Sam to Sarah and vice versa. This visual representation helps simplify and understand the relationships in the given set.
relations in mathematics
In mathematics, a relation is a way to show a connection or relationship between elements of sets. Relations can help categorize and connect different items based on certain criteria or rules.
This exercise involves a specific type of relation: 'has the same first letter as.' This relation is reflexive, meaning each name is related to itself. It also can be symmetric, meaning if 'Sam' is related to 'Sarah', then 'Sarah' is also related to 'Sam'. The relation might not always be transitive, which usually means if 'A' is related to 'B', and 'B' is related to 'C', then 'A' must be related to 'C'.
This particular problem uses the concept of reflexivity and symmetry to create a visual digraph.
step-by-step problem solving
Step-by-step problem solving in mathematics involves breaking down a problem into smaller, manageable parts and solving each part one at a time.
In this exercise, drawing a digraph for the relation involves several clear steps.

1. **Understand the Relation:** Define what the relation means in simple terms. For instance, here, the relation is 'having the same first letter.'
2. **Identify Key Elements:** List out the elements to see who can be related to whom. Here, you write down the names and their starting letters.
3. **Group by Relation:** Organize the names according to the relation. Group names with the same first letter.
4. **Draw Vertices:** Create a vertex for each name on the graph.
5. **Create Directed Edges:** Draw the arrows (edges) between names that share the first letter.
6. **Verify Your Work:** Check to ensure all relations are properly represented in the digraph.
Following these steps ensures accuracy and clarity when tackling similar mathematical problems.

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Most popular questions from this chapter

Now suppose again we are choosing three distinct flavors of ice cream out of four, but instead of putting scoops in a cone or choosing pints, we are going to have the three scoops arranged symmetrically in a circular dish. Similarly to choosing three pints, we can describe a selection of ice cream in terms of which one goes in the dish first, which one goes in second (say to the right of the first), and which one goes in third (say to the right of the second scoop, which makes it to the left of the first scoop). But again, two of these lists will sometimes be equivalent. Once they are in the dish, we can't tell which one went in first. However, there is a subtle difference between putting each flavor in its own small dish and putting all three flavors in a circle in a larger dish. Think about what makes the lists of flavors equivalent, and draw the directed graph whose vertices consist of all lists of three of the flavors of ice cream and whose edges connect two lists that we cannot tell the difference between as dishes of ice cream. How many dishes of ice cream can we distinguish from one another? (h)

Find the cycles of the permutation $$ \left(\begin{array}{ccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 4 & 6 & 2 & 9 & 7 & 1 & 5 & 8 \end{array}\right) . $$

Suppose we make a necklace by stringing 12 pieces of brightly colored plastic tubing onto a string and fastening the ends of the string together. We have ample supplies blue, green, red, and yellow tubing available. Give a generating function in which the coefficient of \(B^{i} G^{j} R^{k} Y^{h}\) is the number of necklaces we can make with \(i\) blues, \(j\) greens, \(k\) reds, and \(h\) yellows. How many terms would this generating function have if you expanded it in terms of powers of \(B, G, R,\) and \(Y ?\) Does it make sense to do this expansion? How many of these necklaces have 3 blues, 3 greens, 2 reds, and 4 yellows?

Given the partition \\{1,3\\},\\{2,4,6\\},\\{5\\} of the set \(\\{1,2,3,4,5,6\\},\) define two elements of \\{1,2,3,4,5,6\\} to be related if they are in the same part of the partition. That is, define 1 to be related to 3 (and 1 and 3 each related to itself), define 2 and 4,2 and \(6,\) and 4 and 6 to be related (and each of \(2,4,\) and 6 to be related to itself), and define 5 to be related to itself. Show that this relation is an equivalence relation.

Draw the digraph of the relation from the set \\{Sam, Mary, Pat, Ann, Polly, Sarah \(\\}\) to the set \(\\{\mathrm{A}, \mathrm{M}, \mathrm{P}, \mathrm{S}\\}\) given by "has as its first letter."
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