Math Problem Statement

1 0 1 0 ​

0 0 0 0 ​

1 0 1 0 ​

0 0 0 1 ​

is this matrix transitive

Solution

To determine whether the given matrix is transitive, we need to verify if it satisfies the transitivity property of a relation.

Transitivity Property:

A relation RR on a set is transitive if, whenever (a,b)R(a, b) \in R and (b,c)R(b, c) \in R, then (a,c)R(a, c) \in R.

The given matrix is:

1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$$ This is a **4x4** matrix, which represents a relation on a set of 4 elements. The entry in row $$i$$ and column $$j$$ is 1 if there is a relation between element $$i$$ and element $$j$$, and 0 otherwise. ### Steps to check for transitivity: 1. For each pair $$(i, j)$$ where $$M[i][j] = 1$$, check if for all $$k$$, whenever $$M[j][k] = 1$$, we have $$M[i][k] = 1$$ as well. - **Row 1**: $$(1, 1) = 1$$, $$(1, 3) = 1$$ - Check if these pairs are transitive: - For $$(1, 1)$$, check all possible pairs with row 1. Since row 1 does not have further links, it's fine. - For $$(1, 3)$$, check if $$M[3][k]$$ holds transitivity. Since $$M[3][3] = 1$$, we must check if $$M[1][3] = 1$$ (which is true), so it holds. - **Row 2**: All entries are zero, so no transitivity violations here. - **Row 3**: $$(3, 1) = 1$$, $$(3, 3) = 1$$ - Check: - For $$(3, 1)$$, since row 1 does not have further relations, it's fine. - For $$(3, 3)$$, check transitivity. Similar to row 1, no violations. - **Row 4**: $$(4, 4) = 1$$, no further links, so no issues. ### Conclusion: Since no violations of the transitivity property were found, **the given matrix is transitive**. Would you like more details or have any questions? Here are some related questions for deeper understanding: 1. What are other properties like reflexivity and symmetry in relations? 2. How do we check if a matrix represents a reflexive relation? 3. How would the matrix change if it were not transitive? 4. What are some examples of non-transitive relations? 5. How is matrix multiplication related to transitivity in relations? **Tip**: When checking for transitivity in matrices, think of multiplying the matrix with itself to see if the relation holds for paths longer than one step.

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Math Problem Analysis

Mathematical Concepts

Relations
Transitivity
Matrix Theory

Formulas

-

Theorems

Transitivity Property

Suitable Grade Level

Grades 9-12