We are asked to sketch the set of points defined by given conditions, describe its boundary, and determine whether the set is open, closed, or neither. We will solve problem 27. The set is defined by $\{(x, y): 2 \le x \le 4, 1 \le y \le 5\}$.

1. Problem Description

We are asked to sketch the set of points defined by given conditions, describe its boundary, and determine whether the set is open, closed, or neither. We will solve problem

2

7. The set is defined by $\{(x, y): 2 \le x \le 4, 1 \le y \le 5\}$.

2. Solution Steps

The set is defined by $2 \le x \le 4$ and $1 \le y \le 5$. This represents a rectangle in the $xy$-plane. The vertices of the rectangle are $(2, 1)$, $(4, 1)$, $(4, 5)$, and $(2, 5)$.

The boundary of the set is the set of points $(x, y)$ such that $x = 2$, $2 \le y \le 5$; $x = 4$, $1 \le y \le 5$; $y = 1$, $2 \le x \le 4$; and $y = 5$, $2 \le x \le 4$.

Since the inequalities are non-strict ($\le$), the boundary is included in the set.

A set is closed if it contains all its boundary points. In this case, the set includes its boundary, so the set is closed.

A set is open if every point in the set has a neighborhood contained in the set. Since the boundary points are in the set, and any neighborhood around them will contain points outside the set, the set is not open.

3. Final Answer

The set is a rectangle with vertices (2, 1), (4, 1), (4, 5), and (2, 5).

The boundary is the rectangle's perimeter.

The set is closed.