Boundedness refers to the property of a set in which all elements of the set lie within a certain range or limit. In other words, a bounded set has a finite or limited extent in some dimension or parameter.
In mathematics, the boundedness of a set is determined by considering the values of the elements in the set and whether they fall within a certain range or limit. For example, in the set |x| + |y| <= 2, all possible combinations of x and y values that satisfy the equation fall within a range of -2 to 2.
The equation |x| + |y| <= 2 represents a set of points in a two-dimensional coordinate system that lie within a certain distance from the origin. The distance from the origin is limited to 2 units, resulting in a bounded set.
To prove that the set |x| + |y| <= 2 is bounded, we can use the triangle inequality property, which states that the sum of any two sides of a triangle must be greater than the third side. In this case, the sum of |x| and |y| must be less than or equal to 2, which shows that all points in the set lie within a certain distance from the origin and therefore, the set is bounded.
No, a set cannot be both bounded and unbounded. A set is either bounded, meaning all elements fall within a certain range, or unbounded, meaning there is no limit to the values of the elements. A set cannot have both bounded and unbounded elements.