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 $M_R$:

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".