Problem 7 Determine whether the relations ... [FREE SOLUTION]

Chapter 9: Problem 7

Determine whether the relations represented by these zero鈥搊ne matrices are partial orders. a) \(\left[\begin{array}{lll}{1} & {1} & {1} \\ {1} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]\) b) \(\left[\begin{array}{lll}{1} & {1} & {1} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]\) c) \(\left[\begin{array}{cccc}{1} & {1} & {1} & {0} \\ {0} & {1} & {1} & {0} \\\ {0} & {0} & {1} & {1} \\ {1} & {1} & {0} & {1}\end{array}\right]\)

Short Answer

Expert verified

a) Yes, b) Yes, c) No.

Step by step solution

- Identify the properties of a partial order

A relation is a partial order if it is reflexive, antisymmetric, and transitive. Begin by checking these properties for each matrix.

- Check reflexivity for part (a)

A relation is reflexive if the diagonal elements are all 1. For matrix (a): \[\begin{array}{lll}{1} & {1} & {1} \ {1} & {1} & {0} \ {0} & {0} & {1}\thend{array}\] The diagonal elements are all 1, hence reflexive.

- Check antisymmetry for part (a)

A relation is antisymmetric if \(a_{ij} = 1\) and \(a_{ji} = 1\) implies \(i = j\). In matrix (a), there are no pairs (i, j) where both \(a_{ij}\) and \(a_{ji}\) are 1 unless \(i = j\). Hence, it is antisymmetric.

- Check transitivity for part (a)

A relation is transitive if whenever \(a_{ij} = 1\) and \(a_{jk} = 1\), \(a_{ik} = 1\). For matrix (a), verify all such conditions: (1,1) and (1,1) gives (1,1), (1,1) and (1,3) gives (1,3), etc. It holds for all, making it transitive.

- Compile part (a)

Since matrix (a) is reflexive, antisymmetric, and transitive, it is a partial order.

- Check reflexivity for part (b)

For matrix (b): \[ \begin{array}{lll}{1} & {1} & {1} \ {0} & {1} & {0} \ {0} & {0} & {1}\thend{array}\] The diagonal elements are all 1, hence reflexive.

- Check antisymmetry for part (b)

Verify antisymmetry on matrix (b). There are no (i, j) pairs where both \(a_{ij} = 1\) and \(a_{ji} = 1\). Therefore, it is antisymmetric.

- Check transitivity for part (b)

Check transitivity on matrix (b). For instance, (1,2) and (2,3) would require (1,3) = 1. Since this holds for all pairs, it is transitive.

- Compile part (b)

Since matrix (b) is reflexive, antisymmetric, and transitive, it too is a partial order.

- Check reflexivity for part (c)

For matrix (c): \[\begin{array}{cccc}{1} & {1} & {1} & {0} \ {0} & {1} & {1} & {0} \ {0} & {0} & {1} & {1} \ {1} & {1} & {0} & {1}\thend{array}\] The diagonal elements are all 1, hence reflexive.

- Check antisymmetry for part (c)

Verify antisymmetry on matrix (c). Here, \(a_{14} = 0\) while \(a_{41} = 1\), violating antisymmetry. Hence, it is not a partial order.

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Key Concepts

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

Reflexivity

Reflexivity is a critical property of partial orders. A relation is reflexive if every element is related to itself. This is represented in a zero-one matrix by having all the diagonal elements equal to 1. Essentially, for an element to be reflexive in matrix form, every position \(a_{ii}\) should be 1. This tells us that each element is always related to itself.

Antisymmetry

Antisymmetry refers to the condition where \(a_{ij} = 1\) and \(a_{ji} = 1\) imply \(i = j\). In simpler terms, if one element is related to another and vice versa, they must be the same element. This property ensures that there are no cycles in the relation except self-loops. Within a zero-one matrix, antisymmetry means no off-diagonal pairs \((i, j)\) and \((j, i)\) both equal 1.

Transitivity

Transitivity is another crucial property for partial orders. A relation is transitive if whenever \(a_{ij} = 1\) and \(a_{jk} = 1\), then \(a_{ik} = 1\). This means if element i is related to j, and j to k, then i must be related to k. For the matrix, you need to check all the paths: if you can reach from i to k via j, then there should be a direct link from i to k.

Zero-One Matrices

Zero-one matrices are used to represent relations in a structured way. Each element in the matrix can be either 0 (indicating no relation) or 1 (indicating a relation). You read these matrices by rows and columns; the element at row i and column j tells whether i is related to j. By inspecting the diagonal, you check for reflexivity, and by looking at off-diagonal pairs for antisymmetry. For transitivity, you review the potential paths.

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