In other math, what does the term “Isometry” refer to?
A) A function preserving only distance between metric spaces.
B) A function that is both injective and surjective.
C) A function preserving operations between metric spaces.
D) A function altering the metric structure of spaces.
The correct answer and explanation is:
The correct answer is A) A function preserving only distance between metric spaces.
Explanation:
In mathematics, an isometry is a type of function between metric spaces that preserves distances. This means that the distance between any two points in the first space is exactly the same as the distance between their images in the second space under the isometry. In other words, if you have two points, xx and yy, in a metric space XX, and an isometry f:X→Yf: X \to Y that maps those points to f(x)f(x) and f(y)f(y) in another metric space YY, then the distance between xx and yy in space XX is the same as the distance between f(x)f(x) and f(y)f(y) in space YY. Mathematically, this can be expressed as: dX(x,y)=dY(f(x),f(y))d_X(x, y) = d_Y(f(x), f(y))
Where dXd_X and dYd_Y are the distance functions (or metrics) in the respective spaces.
This property makes isometries particularly important in geometry, as they help study the invariance of shapes and distances when transformed. Common examples of isometries include translations, rotations, and reflections in Euclidean space. These transformations do not change the relative distances or angles between points, hence preserving the structure of the space.
Isometries can be thought of as distance-preserving “motions” that keep the spatial relationships intact, which is crucial in both theoretical and applied mathematics.
To summarize, isometries are functions that preserve the distances between points, making them central to understanding and working with metric spaces and geometric objects.