Question

Describe geometrically the following sets in the \(x y\) -plane. (i) \(\\{(x, y) \mid xx^{2}\right\\}\) (v) \(\\{(x, y)|| x|+| y \mid<4\\} ;\) (vi) \(\left\\{(x, y) \mid(x-2)^{2}+(y+5)^{2} \leq 9\right\\}\) (vii) \(\\{(x, y) \mid x=0\\} ;\) (viii) \(\left\\{(x, y) \mid x^{2}-2 x y+y^{2}<0\right\\} ;\) (ix) \(\left\\{(x, y) \mid x^{2}-2 x y+y^{2}=0\right\\}\).

Short answer

(i) Open half-plane above \(y=x\); (ii) Open disk inside unit circle; (iii) Open square with \(|x|,|y|<1\); (iv) Region above \(y=x^2\); (v) Open diamond \(|x|+|y|<4\); (vi) Closed disk, center \((2,-5)\), radius 3; (vii) Line \(x=0\); (viii) Empty set; (ix) Line \(y=x\).

Step-by-step solution

Interpret Set (i)

The inequality \(x < y\) represents a collection of points in the plane where the coordinate \(x\) is less than the coordinate \(y\). Geometrically, this is the area above the line \(y = x\), but not including the line itself. It is an open half-plane.

Interpret Set (ii)

The inequality \(x^2 + y^2 < 1\) describes all the points that are located inside the circle centered at the origin with radius 1. This region does not include the circle itself, forming an open disk.

Interpret Set (iii)

The set described by \(\max(|x|,|y|)<1\) defines a square centered at the origin with vertices at \((1,1)\), \((-1,1)\), \((-1,-1)\), and \((1,-1)\). This square is open, meaning it does not include its boundary line segments.

Interpret Set (iv)

The inequality \(y > x^2\) represents the area above the parabola \(y = x^2\). This parabola opens upwards from the vertex at (0,0), forming a region above it but not including the parabola itself.

Interpret Set (v)

The expression \(|x| + |y| < 4\) describes the region inside a diamond-shaped area (a rhombus) centered at the origin with vertices at \((4,0)\), \((0,4)\), \((-4,0)\), and \((0,-4)\). This region is open, excluding the boundary.

Interpret Set (vi)

The inequality \((x-2)^2 + (y+5)^2 \leq 9\) represents a closed disk centered at \((2, -5)\) with radius 3. The inclusive inequality means that the boundary circle is included.

Interpret Set (vii)

The condition \(x=0\) corresponds to the vertical line passing through the origin. This line includes all the points where \(x\) is zero, spanning all possible \(y\) values.

Interpret Set (viii)

The expression \(x^2 - 2xy + y^2 < 0\) simplifies to \((x-y)^2 < 0\), which is not possible for any real \(x\) and \(y\). So, this set contains no real points in the plane.

Interpret Set (ix)

The expression \(x^2 - 2xy + y^2 = 0\) simplifies to \((x-y)^2 = 0\), meaning \(x = y\). This is the line \(y = x\) on the plane.

Key concepts

Inequalities in the Coordinate Plane

When dealing with inequalities in the coordinate plane, we are thinking about regions rather than specific points or lines. An inequality such as \(x < y\) describes the area in the plane where the x-coordinate is less than the y-coordinate. This inequality corresponds to a region above a certain line, in this case, the line \(y = x\) but does not include the line itself, resulting in what is known as an 'open half-plane.' Such an inequality focuses on a condition that creates a full set of points forming a particular spatial area.

An example from above, \(x^2 + y^2 < 1\), describes a set of inequalities where points fall inside of a circle. It ensures that distances from the origin to each point remain less than 1. Inequalities, therefore, help us define specific areas or regions in the plane — sometimes forming sophisticated shapes based on conditions applied to the coordinates \(x\) and \(y\).

Open and Closed Regions

Open and closed regions refer to whether the boundary of a set is included (closed) or not included (open) in the region described. For example, the inequality \((x-2)^2 + (y+5)^2 \leq 9\) represents a closed region because the 'equal to' part of the inequality means that all points on the boundary circle are also included, forming a closed disk.

In contrast, an open region like the one described by \(y > x^2\), represents a space above the parabola where none of the boundary points on the parabola itself are part of the region. Such distinctions are crucial when interpreting graphs because they define the exact extent of regions on a plane.

• Open regions do not include boundaries.
• Closed regions include boundary lines.

Graphical Representation of Sets

The graphical representation of sets on the coordinate plane allows us to visualize abstract mathematical concepts easily. Each set or inequality corresponds to a specific geometric shape or area. For instance, equations that involve maximums, like \(\max(|x|,|y|)<1\), create interesting shapes such as a square when centered at the origin. The resulting figure demonstrates how the maximum constraint decrees the boundary within which coordinates can exist.

Visualizing these inequalities and equations helps convey the underlying mathematical ideas more intuitively. By plotting an equation like \(x=0\), a simple vertical line is depicted on the graph that visually marks a clear location where all points meet the specified condition. Converting such algebraic sets into visual format empowers students to better grasp relationships and spatial distribution of points within regions dictated by inequalities or equalities.