https://file.aiquickdraw.com/mathbot/file/1727103417304t1jnoj8x.jpg

Let's analyze the matrix provided in the image to determine if the relation it represents is symmetric, antisymmetric, or transitive.

The matrix representing the relation is:

\[
M = \begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]

### Symmetric:
A relation is symmetric if for every pair \((a, b)\), whenever \((a, b)\) is in the relation, \((b, a)\) is also in the relation. In terms of the matrix, this means the matrix should be equal to its transpose, i.e., \( M = M^T \).

Checking if the matrix is symmetric:

\[
M^T = \begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]

Since \(M = M^T\), the relation is symmetric.

### Antisymmetric:
A relation is antisymmetric if for all pairs \((a, b)\), if both \((a, b)\) and \((b, a)\) are in the relation, then \(a = b\). This means the matrix should not have both \(M_{ij} = 1\) and \(M_{ji} = 1\) unless \(i = j\).

Looking at the matrix, we observe that:
- \(M_{13} = 1\) and \(M_{31} = 1\), but \(1 \neq 3\).

This violates the antisymmetry condition. Therefore, **the relation is not antisymmetric**.

### Transitive:
A relation is transitive if whenever \((a, b)\) and \((b, c)\) are in the relation, then \((a, c)\) must also be in the relation. To check this in the matrix, for all pairs where \(M_{ij} = 1\) and \(M_{jk} = 1\), we need to ensure that \(M_{ik} = 1\).

- \(M_{13} = 1\) and \(M_{33} = 1\), but \(M_{13} = 1\), so this is consistent.
- There are no other pairs that create a violation.

Thus, the relation is **transitive**.

### Summary:
- **Symmetric**: Yes
- **Antisymmetric**: No
- **Transitive**: Yes

Do you have any questions or want further details?

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Here are some related questions you might want to consider:
1. What does it mean for a relation to be reflexive, and is this matrix reflexive?
2. How would the matrix change if we wanted to make the relation antisymmetric?
3. Can a relation be both symmetric and antisymmetric? Under what conditions?
4. How does transitivity relate to the composition of relations?
5. How would you represent a reflexive, symmetric, and transitive relation (an equivalence relation) using a matrix?

**Tip**: When analyzing relations using matrices, always check the diagonal for reflexivity and use symmetry properties to infer antisymmetry.

Find whether the relation represented by the matrix [1 0 1 0, 0 0 0 0, 1 0 1 0, 0 0 0 1] is symmetric, antisymmetric, and transitive.

Explore whether the matrix [1 0 1 0, 0 0 0 0, 1 0 1 0, 0 0 0 1] satisfies symmetry, antisymmetry, and transitivity properties. Learn about key concepts such as matrix transposition, antisymmetry conditions, and transitivity in relation theory.

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