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Chapter 9: Problem 29

What is the covering relation of the partial ordering \(\\{(A, B) | A \subseteq B\\}\) on the power set of \(S,\) where \(S=\\{a, b, c\\} ?\)

Short Answer

Expert verified
The covering relations are pairs of subsets that differ by exactly one element, such as \((\emptyset, \{a\})\) and \((\{a\}, \{a, b\})\).

Step by step solution

01

Understand the Power Set

The power set of set \(S = \{a, b, c\}\) includes all possible subsets of \(S\). These subsets are: \(\emptyset, \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\).
02

Define the Partial Ordering

The partial ordering relation given is \(\{(A, B) | A \,\subseteq\, B\}\). This means for any two subsets \(A\) and \(B\) in the power set, \(A\) is related to \(B\) if and only if \(A\) is a subset of \(B\).
03

Determine the Covering Relations

The covering relation \(A \,\lessdot\, B\) holds if \(A \,\subseteq\, B\) and there is no subset \(C\) such that \(A \,\subseteq\, C\) and \(C \,\subseteq\, B\) where \(C \,eq A\) and \(C \,eq B\). Essentially, \(B\) is covered by \(A\) if \(B\) differs from \(A\) by exactly one element.
04

List the Covering Relations

By considering the definition in Step 3, the covering relations in the power set of \(S\) are: \((\emptyset, \{a\}), (\emptyset, \{b\}), (\emptyset, \{c\}), (\{a\}, \{a, b\}), (\{a\}, \{a, c\}), (\{b\}, \{a, b\}), (\{b\}, \{b, c\}), (\{c\}, \{a, c\}), (\{c\}, \{b, c\}), (\{a, b\}, \{a, b, c\}), (\{a, c\}, \{a, b, c\}), (\{b, c\}, \{a, b, c\})\).

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Key Concepts

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

Power Set
The power set of a set is a fundamental concept in discrete mathematics. It involves all possible subsets of a given set.
Let's illustrate this with an example.
Consider set \(S = \{a, b, c\}\). To find the power set of \(S\), we need to list every combination of elements that can be found in \(S\).

These include:
  • The empty set \(\emptyset\)
  • Single element sets: \(\{a\}, \{b\}, \{c\}\)
  • Two-element subsets: \(\{a, b\}, \{a, c\}, \{b, c\}\)
  • The full set: \(\{a, b, c\}\)
All these subsets collectively form the Power Set of \(S\). For any set \(S\) with \(n\) elements, the power set of \(S\) will have \(2^n\) elements. In our example, since \(S\) has 3 elements, its power set will consist of \(2^3 = 8\) subsets.
Subset Relation
The subset relation is a key concept in understanding partial orderings.
Whenever we say that set \(A\) is a subset of set \(B\) (written as \(A \subseteq B\)), we mean every element of \(A\) is also an element of \(B\).
Here's a breakdown:
  • If \(A = \{a\}\) and \(B = \{a, b\}\), then \(A \subseteq B\) because 'a' is in both \(A\) and \(B\).
  • However, if \(A = \{a, d\}\) and \(B = \{a, b\}\), then \(A ot\subseteq B\) because 'd' is not in \(B\).
Partial orderings, where we use subsets to define relations, help us clearly understand hierarchical structures.
Discrete Mathematics
Discrete mathematics is a branch of mathematics dealing with distinct and separated values. It contrasts with continuous mathematics, which studies processes that change smoothly.
Key areas include:
  • Set theory
  • Graph theory
  • Mathematical logic
  • Combinatorics
  • Algorithms
Each of these areas involves concepts that are fundamental for computer science, cryptography, and optimization problems.
Understanding discrete mathematics is crucial for effectively dealing with problems in these areas, such as creating algorithms, proving the correctness of logical statements, and modeling network structures.

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Most popular questions from this chapter

Which of these are posets? a) \((\mathbf{R},=)\) b) \((\mathbf{R},<)\) c) \((\mathbf{R}, \leq)\) d) \((\mathbf{R}, \neq)\)

Let \(A\) be the set of students at your school and \(B\) the set of books in the school library. Let \(R_{1}\) and \(R_{2}\) be the relations consisting of all ordered pairs \((a, b),\) where student \(a\) is required to read book \(b\) in a course, and where student \(a\) is required to read book \(b\) in a course, and where student \(a\) has read book \(b\) , respectively. Describe the ordered pairs in each of these relations. $$ \begin{array}{ll}{\text { a) } R_{1} \cup R_{2}} & {\text { b) } R_{1} \cap R_{2}} \\ {\text { c) } R_{1} \oplus R_{2}} & {\text { d) } R_{1}-R_{2}} \\\ {\text { e) } R_{2}-R_{1}}\end{array} $$

Determine whether the relations represented by these zero-one matrices are equivalence relations. $$ \text { a) }\left[\begin{array}{lll}{1} & {1} & {1} \\ {0} & {1} & {1} \\\ {1} & {1} & {1}\end{array}\right] \quad \text { b) }\left[\begin{array}{llll}{1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1} \\\ {1} & {0} & {1} & {0} \\ {0} & {1} & {0} & {1}\end{array}\right] \text { c) }\left[\begin{array}{llll}{1} & {1} & {1} & {0} \\ {1} & {1} & {1} & {0} \\\ {1} & {1} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right] $$

a) How many relations are there on the set \(\\{a, b, c, d\\} ?\) b) How many relations are there on the set \(\\{a, b, c, d\\}\) that contain the pair \((a, a) ?\)

a) Show that there is exactly one maximal element in a poset with a greatest element. b) Show that there is exactly one minimal element in a poset with a least element.
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