Let's break down the question step by step.

We have a set $A = \{1, 2, 3\}$, and a binary relation $R$ on $A$, defined as:

$R = \{ (a, b) \in A \times A \mid a \leq b \}$

This means the relation includes all pairs where the first element $a$ is less than or equal to the second element $b$.

We need to determine which of the matrices represents this relation.

Steps to find the correct matrix:

1. We will represent each element of the set $A = \{1, 2, 3\}$ by its index in a matrix, i.e., $1$ is represented by the first row/column, $2$ by the second row/column, and $3$ by the third row/column.
2. For each pair $(a, b)$, we set the entry in the matrix to 1 if $a \leq b$, otherwise 0.
   • Row 1 corresponds to $a = 1$, and it should have 1's in columns where $1 \leq b$, i.e., columns for $b = 1, 2, 3$.
   • Row 2 corresponds to $a = 2$, and it should have 1's in columns where $2 \leq b$, i.e., columns for $b = 2, 3$.
   • Row 3 corresponds to $a = 3$, and it should have 1's in columns where $3 \leq b$, i.e., columns for $b = 3$.

Let's now check the provided options. Based on this reasoning, I'll determine which matrix corresponds to the relation $R$. Let me check the image for details. PIL import Image

Open the uploaded image to analyze its contents

img_path = '/mnt/data/1738902298976fb8iijz4.jpg' img = Image.open(img_path) img.show()I have opened the image. Now I will analyze the matrices and their corresponding relationships based on the criteria we discussed. Based on the set $A = \{1, 2, 3\}$ and the relation $R = \{(a, b) \mid a \leq b\}$, let's check which of the matrices matches this relation.

Given the conditions for each row in the matrix based on $a \leq b$, here is the correct pattern:

• Row 1: Since $1 \leq 1, 1 \leq 2, 1 \leq 3$, all entries in row 1 should be 1.
• Row 2: Since $2 \leq 2, 2 \leq 3$, entries in row 2 for columns 2 and 3 should be 1.
• Row 3: Since $3 \leq 3$, only the entry in row 3, column 3 should be 1.

This leads to the correct matrix: