To write the relation $R$ as a table where $(x, y) \in R$ if and only if $x^2 \geq y$, let's first understand the condition:

The set $A = \{1, 2, 3, 4\}$ represents the elements involved in the relation. The relation condition states that for each pair $(x, y)$, $x^2 \geq y$ must hold true. This means for each value of $x$, we need to check whether $x^2$ is greater than or equal to $y$, for all $y \in \{1, 2, 3, 4\}$.

Let’s calculate $x^2$ for each $x \in \{1, 2, 3, 4\}$:

• For $x = 1$, $1^2 = 1$
• For $x = 2$, $2^2 = 4$
• For $x = 3$, $3^2 = 9$
• For $x = 4$, $4^2 = 16$

Now, for each $x$, check if $x^2 \geq y$:

1. For $x = 1$:

   • $1^2 \geq 1$, true (so $(1, 1)$ is in $R$)
   • $1^2 \geq 2$, false
   • $1^2 \geq 3$, false
   • $1^2 \geq 4$, false

   So, the pairs for $x = 1$ are $(1, 1)$.

2. For $x = 2$:

   • $2^2 \geq 1$, true (so $(2, 1)$ is in $R$)
   • $2^2 \geq 2$, true (so $(2, 2)$ is in $R$)
   • $2^2 \geq 3$, true (so $(2, 3)$ is in $R$)
   • $2^2 \geq 4$, true (so $(2, 4)$ is in $R$)

   So, the pairs for $x = 2$ are $(2, 1), (2, 2), (2, 3), (2, 4)$.

3. For $x = 3$:

   • $3^2 \geq 1$, true (so $(3, 1)$ is in $R$)
   • $3^2 \geq 2$, true (so $(3, 2)$ is in $R$)
   • $3^2 \geq 3$, true (so $(3, 3)$ is in $R$)
   • $3^2 \geq 4$, true (so $(3, 4)$ is in $R$)

   So, the pairs for $x = 3$ are $(3, 1), (3, 2), (3, 3), (3, 4)$.

4. For $x = 4$:

   • $4^2 \geq 1$, true (so $(4, 1)$ is in $R$)
   • $4^2 \geq 2$, true (so $(4, 2)$ is in $R$)
   • $4^2 \geq 3$, true (so $(4, 3)$ is in $R$)
   • $4^2 \geq 4$, true (so $(4, 4)$ is in $R$)

   So, the pairs for $x = 4$ are $(4, 1), (4, 2), (4, 3), (4, 4)$.

Now, let's put this information into a table format. The relation $R$ is:

x \ y	1	2	3	4
1	✓
2	✓	✓	✓	✓
3	✓	✓	✓	✓
4	✓	✓	✓	✓

Where "✓" indicates that $(x, y) \in R$.

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Would you like further details or clarifications on any part? Here are a few related questions to explore:

1. How can you describe the properties of the relation $R$?
2. What happens if we change the condition to $x^2 < y$?
3. How can you represent a relation using set notation?
4. Can a relation be symmetric in this case? Why or why not?
5. How can you test if a relation is transitive using this table?

Tip: For relations, the table format can be very useful for visualizing which pairs satisfy the given condition!