4. Sketch the sets \( X=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2} \leq 1\right\} \) and \( Y=\left\{(x, y) \in \mathbb{R}^{2} \left\lvert\, x-\frac{1}{2} \leq y \leq x+\frac{1}{2}\right.\right\} \) on \( \mathbb{R}^{2} \). On separate drawings, shade the sets \( X \cup Y, X \cap Y, X-Y \), and \( Y-X \).
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To sketch the provided sets clearly, let’s break down the descriptions of both \( X \) and \( Y \): **First Set \( X \):** The set \( X \) represents all points \((x,y)\) within or on the boundary of a circle centered at the origin \((0,0)\) with a radius of 1. This includes all points that satisfy the equation \( x^2 + y^2 \leq 1 \). When you sketch this set, you will draw a filled circle with radius 1, covering the entire area inside the circle. **Second Set \( Y \):** The set \( Y \) defines a region between two lines: \( y = x - \frac{1}{2} \) and \( y = x + \frac{1}{2} \). To visualize this, you first plot these two lines on your Cartesian plane. The line \( y = x - \frac{1}{2} \) has a slope of 1 and intercepts the y-axis at -0.5, while the line \( y = x + \frac{1}{2} \) also has a slope of 1 but intercepts at +0.5. The area between these lines is shaded, representing all points satisfying \( x - \frac{1}{2} \leq y \leq x + \frac{1}{2} \). Now, let’s tackle the combined sketches: 1. **For \( X \cup Y \):** Combine the filled circle from set \( X \) and the strip area from \( Y \). Shade the entire area that belongs to either \( X \) or \( Y \). The resulting sketch will depict the full circle with the additional strip of points lying between the two lines overlapping as necessary. 2. **For \( X \cap Y \):** Shade only the area where the circle overlaps with the strip of lines defined by \( Y \). Conceptually, this is where you take the intersection of both sets: only the parts of the circle that lie within the boundaries created by the lines of \( Y \). 3. **For \( X - Y \):** In this case, you want to shade the parts of the circle that do not intersect with the area defined by \( Y \). Essentially, any area of the circle that falls outside the strip of lines should be shaded. 4. **For \( Y - X \):** This involves shading the area within the strip defined by \( Y \) that does not overlap with the circle \( X \). You draw this by identifying the points in \( Y \) that are entirely outside of the circle’s perimeter. By organizing these sketches carefully, you'll get a comprehensive visual representation of union, intersection, and set differences! Happy sketching!
