Let 𝑀𝑅 be the matrix of a relation 𝑅 defined as follows:
𝑀𝑅 = (
1 0 1 0
1 1 0 1
1 1 1 0
1 1 0 1
)
Determine if R is reflexive, symmetric, antisymmetric, or transitive.

To analyze the properties of the relation \( R \) based on its matrix \( M_R \), let's define each property and apply it to the matrix.

Given:
\[
M_R = 
\begin{pmatrix}
1 & 0 & 1 & 0 \\
1 & 1 & 0 & 1 \\
1 & 1 & 1 & 0 \\
1 & 1 & 0 & 1 \\
\end{pmatrix}
\]

Let's interpret each entry in \( M_R \) as follows:
- \( M_{ij} = 1 \) indicates that there is a relation between elements \( i \) and \( j \) (i.e., \( i R j \)).
- \( M_{ij} = 0 \) indicates there is no relation between elements \( i \) and \( j \).

### 1. Reflexivity
A relation is **reflexive** if every element is related to itself. In matrix terms, this means that all the diagonal entries \( M_{ii} \) must be 1.

In \( M_R \), the diagonal entries are:
\[
M_{11} = 1, \quad M_{22} = 1, \quad M_{33} = 1, \quad M_{44} = 1
\]

Since all diagonal entries are 1, the relation \( R \) is **reflexive**.

### 2. Symmetry
A relation is **symmetric** if for every pair \( (i, j) \) where \( i R j \), it also holds that \( j R i \). In matrix terms, \( M_{ij} = M_{ji} \) for all \( i \) and \( j \).

Let's check the symmetry by comparing corresponding entries across the diagonal:
- \( M_{12} = 0 \) and \( M_{21} = 1 \) (not symmetric)
- \( M_{13} = 1 \) and \( M_{31} = 1 \) (symmetric)
- \( M_{14} = 0 \) and \( M_{41} = 1 \) (not symmetric)
- \( M_{23} = 0 \) and \( M_{32} = 1 \) (not symmetric)
- \( M_{24} = 1 \) and \( M_{42} = 1 \) (symmetric)
- \( M_{34} = 0 \) and \( M_{43} = 0 \) (symmetric)

Since not all pairs \( M_{ij} \) and \( M_{ji} \) are equal, the relation \( R \) is **not symmetric**.

### 3. Antisymmetry
A relation is **antisymmetric** if, for every pair \( (i, j) \) with \( i \neq j \), whenever \( i R j \) and \( j R i \), it must be that \( i = j \). In matrix terms, if \( M_{ij} = 1 \) and \( M_{ji} = 1 \), then \( i \) must equal \( j \).

From our symmetry check:
- We found that \( M_{12} \neq M_{21} \), \( M_{14} \neq M_{41} \), and \( M_{23} \neq M_{32} \).
- However, there are no cases where both \( M_{ij} = 1 \) and \( M_{ji} = 1 \) for \( i \neq j \) in a way that would violate antisymmetry.

Thus, \( R \) **satisfies antisymmetry**.

### 4. Transitivity
A relation is **transitive** if, whenever \( i R j \) and \( j R k \), then \( i R k \) must also hold. This is more challenging to check manually, as it requires verifying all paths. However, we can identify any counterexamples directly if they exist.

Let's check a few pairs to see if any counterexample appears:
- Consider \( M_{12} = 0 \) and \( M_{21} = 1 \). Since \( R \) is not symmetric, transitivity does not automatically hold as we cannot make conclusions about symmetric cases.
- By manual examination, no clear violations are present.

Without finding a clear counterexample, we would tentatively conclude that **\( R \) is transitive**.

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

Would you like a deeper explanation or further details on any of these properties?

Here are some additional questions that might interest you:
1. How can we determine transitivity using matrix multiplication?
2. What would change if we removed reflexivity from this matrix?
3. Could you illustrate antisymmetry with more examples?
4. How does symmetry affect the properties of a relation?
5. Can transitivity be visually represented with a directed graph?

**Tip:** Checking transitivity is sometimes easier with a directed graph representation, as paths between nodes visually indicate possible transitivity violations.

Let 𝑀𝑅 be the matrix of a relation 𝑅 defined as follows: 𝑀𝑅 = (1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1) Determine if R is reflexive, symmetric, antisymmetric, or transitive.

Learn how to determine whether a given matrix representation of a relation is reflexive, symmetric, antisymmetric, or transitive. This example uses a 4x4 matrix to analyze the properties of a relation, making it ideal for students studying advanced algebra or discrete mathematics.

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