let x=(1,2,3,4) and R=(x,y)/x>y  draw the diagraphs of R and determine the matrix of R also examine whether R is symmetric,antisymmetric or antisymmetic

We are given the set \( X = \{1, 2, 3, 4\} \) and the relation \( R = \{(x, y) \mid x > y \} \).

### 1. **Drawing the Digraph of R**
The relation \( R \) consists of pairs where \( x > y \), so we consider all pairs \( (x, y) \) such that \( x \in X \) and \( y \in X \), and \( x > y \). 

Thus, the pairs in \( R \) are:

\[
R = \{(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)\}
\]

- These pairs can be represented as a digraph. In this digraph:
  - Each element of \( X \) is represented as a node.
  - There is a directed edge from node \( x \) to node \( y \) if \( (x, y) \in R \), which corresponds to \( x > y \).

Here's how the edges are:
- From 2 to 1,
- From 3 to 1, and 3 to 2,
- From 4 to 1, 4 to 2, and 4 to 3.

### 2. **Matrix of the Relation \( R \)**

The relation matrix for \( R \) is constructed as a 4x4 matrix (since \( X \) has 4 elements), with rows and columns labeled by the elements of \( X \) in increasing order. An entry \( m_{ij} \) is 1 if \( x_i > x_j \), and 0 otherwise.

For \( X = \{1, 2, 3, 4\} \), the relation \( R \) gives the following matrix \( M_R \):

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

### 3. **Symmetry, Antisymmetry, and Asymmetry**

- **Symmetric:** A relation \( R \) is symmetric if \( (x, y) \in R \implies (y, x) \in R \). In this case, \( R \) is **not symmetric**, since if \( x > y \), then \( y > x \) does not hold.
  
- **Antisymmetric:** A relation \( R \) is antisymmetric if \( (x, y) \in R \) and \( (y, x) \in R \implies x = y \). Since \( (x, y) \in R \) implies \( x > y \), and there are no pairs where both \( (x, y) \in R \) and \( (y, x) \in R \), the relation is **antisymmetric**.

- **Asymmetric:** A relation \( R \) is asymmetric if \( (x, y) \in R \implies (y, x) \notin R \). Since this holds for \( R \) (if \( x > y \), then \( y \not> x \)), the relation is also **asymmetric**.

### Conclusion:

- The digraph of \( R \) consists of directed edges showing the pairs where \( x > y \).
- The matrix of \( R \) is as shown above.
- The relation \( R \) is **antisymmetric** and **asymmetric**, but not **symmetric**.

Would you like more details on any step or have any questions? 

Here are five related questions:
1. What would the relation matrix look like for a different relation, say \( x \geq y \)?
2. How does the property of transitivity apply to this relation?
3. Can you define and give an example of a reflexive relation?
4. What changes in the matrix when the set \( X \) is expanded to more elements?
5. How do you determine if a relation is transitive based on its matrix?

**Tip:** When constructing digraphs, carefully note the direction of arrows, as they represent specific relationships like "greater than" or "less than".

let x = {1, 2, 3, 4} and R = {(x,y) | x > y}  draw the digraphs of R and determine the matrix of R also examine whether R is symmetric, antisymmetric or asymmetric

Explore the relation R defined on the set X = {1, 2, 3, 4} with a detailed analysis of its digraphs and matrix representation. Learn how to determine if the relation is symmetric, antisymmetric, or asymmetric, suitable for Grades 10-12.

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