To sketch the set $X = [1,3] \times [1,2]$, let's break it down:

1. The interval $[1,3]$ on the x-axis includes all x-values between 1 and 3, including the endpoints.
2. The interval $[1,2]$ on the y-axis includes all y-values between 1 and 2, including the endpoints.

Sketching the Set:

• Draw a rectangle on the coordinate plane where:
  • The x-values range from 1 to 3.
  • The y-values range from 1 to 2.
• The corners of this rectangle will be at the points (1, 1), (1, 2), (3, 1), and (3, 2).
• Since the intervals are closed (indicated by the square brackets), the edges of the rectangle are included in the set.

This rectangle represents the Cartesian product $[1,3] \times [1,2]$, which is the set $X$.

Would you like any further details or have any questions about this explanation?

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Here are some related questions to consider:

1. What does the Cartesian product represent in general?
2. How would the sketch change if the intervals were open, e.g., $(1,3) \times (1,2)$?
3. How do you interpret the Cartesian product when one interval is open and the other is closed?
4. What is the area of the set $X$ in the plane?
5. How would you describe the boundary of the set $X$?

Tip: When working with Cartesian products, always consider the type of intervals (open or closed) as they directly affect the boundaries of the set.