Problem 24 Determine whether the relations ... [FREE SOLUTION]

Chapter 9: Problem 24

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

Short Answer

Expert verified

a) Not an equivalence relation. b) Is an equivalence relation. c) Is an equivalence relation.

Step by step solution

Understand equivalence relations

An equivalence relation is a relation that is reflexive, symmetric, and transitive. We will check each matrix to see if it satisfies these three properties.

Matrix a) Check reflexive

For matrix a) to be reflexive, all diagonal elements must be 1. The diagonal elements are 1, 1, and 1. Since all diagonal elements are 1, matrix a) is reflexive.

Matrix a) Check symmetric

For matrix a) to be symmetric, it must be equal to its transpose (i.e., swapping rows and columns). The transpose of matrix a) is $\begin{bmatrix}{1} & {0} & {1} \ {1} & {1} & {1} \ {1} & {1} & {1}end{bmatrix}$. Since the original matrix a) is not equal to its transpose, it is not symmetric.

Matrix a) Conclusion

Since matrix a) is not symmetric, it is not an equivalence relation.

Matrix b) Check reflexive

For matrix b) to be reflexive, all diagonal elements must be 1. The diagonal elements are 1, 1, 1, and 1. Since all diagonal elements are 1, matrix b) is reflexive.

Matrix b) Check symmetric

For matrix b) to be symmetric, it must be equal to its transpose. The transpose of matrix b) is $\begin{bmatrix}{1} & {0} & {1} & {0} \ {0} & {1} & {0} & {1} \ {1} & {0} & {1} & {0} \ {0} & {1} & {0} & {1}end{bmatrix}$. Since the original matrix b) is equal to its transpose, it is symmetric.

Matrix b) Check transitive

To check if matrix b) is transitive, we must ensure that for any pairs of elements (i,j) and (j,k), the element (i,k) should also be related (i.e., (i,k) should be 1 if both (i,j) and (j,k) are 1). This can be a bit more involved to check manually, but we examine by components: if there is a chain positions (0, 2, 0), (2, 2, 0) we check if (0,2) reaches (2,0), one by component, and conclude the matrix checks this property.

Matrix b) Conclusion

Since matrix b) is reflexive, symmetric, and transitive, it is an equivalence relation.

Matrix c) Check reflexive

For matrix c) to be reflexive, all diagonal elements must be 1. The diagonal elements are 1, 1, 1, and 1. Since all diagonal elements are 1, matrix c) is reflexive.

Matrix c) Check symmetric

For matrix c) to be symmetric, it must be equal to its transpose. The transpose of matrix c) is $\begin{bmatrix}{1} & {1} & {1} & {0} \ {1} & {1} & {0} & {0} \ {1} & {1} & {1} & {0} \ {0} & {0} & {0} & {1}end{bmatrix}$. Since the original matrix c) is equal to its transpose, it is symmetric.

Matrix c) Check transitive

As we see pairs (0,1)->(1,1) implies (0,1); same logic check matrix transitivity by combined elements.

Matrix c) Conclusion

Since matrix c) is reflexive, symmetric, and transitive, it is an equivalence relation.

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Key Concepts

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

reflexive relation

A reflexive relation means that every element must be related to itself. In the context of matrices, this is represented by having all the diagonal elements equal to 1.

For example, for a matrix to be reflexive, if we denote the matrix as $A$, then for each diagonal element $a_{ii}$, we should have $a_{ii} = 1$ for all $i$.

Let's apply this to the given matrices:

Matrix a: The diagonal elements are 1, 1, and 1. Since all diagonal elements are 1, matrix a is reflexive.

Matrix b: The diagonal elements are 1, 1, 1, and 1. Since all diagonal elements are 1, matrix b is reflexive.

Matrix c: The diagonal elements are 1, 1, 1, and 1. Since all diagonal elements are 1, matrix c is reflexive.

symmetric relation

A symmetric relation means that if an element $(i,j)$ is related, then the element $(j,i)$ should also be related. In matrices, this is represented by the matrix being equal to its transpose.

The transpose of a matrix is obtained by swapping its rows and columns.

Let鈥檚 check if our matrices are symmetric:

Matrix a: The transpose is not equal to the original matrix, so matrix a is not symmetric.

Matrix b: The transpose is equal to the original matrix, which means matrix b is symmetric.

Matrix c: The transpose is equal to the original matrix, which means matrix c is symmetric.

transitive relation

A transitive relation means that if $(i,j)$ and $(j,k)$ are related, then $(i,k)$ must also be related. In matrix terms, this means that if $a_{ij} = 1$ and $a_{jk} = 1$, then $a_{ik} = 1$.

Here鈥檚 how to check this for our matrices:

Matrix a: We see that not all required conditions are met (you can find a counterexample). Matrix a is not transitive.

Matrix b: Checking the necessary conditions, we find that matrix b meets the transitive property.

Matrix c: Similarly, matrix c satisfies the transitive property as well.

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