SUMMARY
The set S = {(x,y): |x| + |y| <= 2} is indeed bounded. The proof involves demonstrating that all points (x, y) within this set are confined within a square defined by the vertices (2,0), (0,2), (-2,0), and (0,-2) in the Cartesian plane. This confirms that the maximum distance from the origin does not exceed 2, establishing boundedness in terms of Euclidean distance.
PREREQUISITES
- Understanding of Euclidean distance in a two-dimensional space.
- Familiarity with the concept of bounded sets in mathematics.
- Basic knowledge of inequalities and their graphical representations.
- Ability to manipulate algebraic expressions involving absolute values.
NEXT STEPS
- Study the properties of bounded sets in metric spaces.
- Learn about the graphical representation of inequalities in two dimensions.
- Explore the implications of the triangle inequality in relation to boundedness.
- Investigate the relationship between absolute values and distance metrics.
USEFUL FOR
Mathematicians, students studying real analysis, and anyone interested in understanding geometric properties of sets in Euclidean spaces.