True or false
The union of two convex sets is convex. (b) the complement of a convex set is convex.
The Correct Answer and Explanation is:
Let’s examine both statements:
(a) The union of two convex sets is convex.
False.
A convex set is defined as a set where, for any two points within the set, the line segment joining them lies entirely within the set. However, the union of two convex sets does not necessarily retain this property.
To see why, let’s consider two convex sets, (A) and (B), which do not overlap. For example, let (A) be a circle centered at the origin and (B) be a circle centered at a point far from the origin. The union of these two circles, (A \cup B), is not convex. If you take one point from (A) and another from (B), the line segment joining these two points will lie outside the union of the two circles, which violates the condition of convexity.
In contrast, if two convex sets intersect, their union may still be convex, but this is not guaranteed for all cases. Therefore, the union of two convex sets is not always convex.
(b) The complement of a convex set is convex.
False.
The complement of a convex set is typically not convex. To understand why, consider a convex set (C) in a space. The complement of (C) is the set of all points that do not belong to (C), i.e., the set (C^c = { x \mid x \notin C }).
For example, if (C) is a convex disk, then the complement (C^c) consists of all points outside the disk. Now, if you take two points inside (C^c), the line segment connecting them may pass through points inside (C), which means the segment is not fully contained in the complement. Hence, (C^c) is not convex.
Convexity of a set is a geometric property that is not preserved when taking complements. The complement of a convex set tends to exhibit “concave” features rather than maintaining the convexity property.
Conclusion:
- The union of two convex sets is not necessarily convex.
- The complement of a convex set is also not convex.
Thus, both statements are false.