Question

Let T: Rⁿ -> Rᵐ be a one-to-one mapping from Rⁿ to Rᵐ then m>=n

Is it true or false? please explain briefly

Answer 1

The statement is true: if we have a one-to-one mapping from R^n to R^m, then it must be the case that m >= n.

Step-by-step explanation:

The statement is true: if we have a one-to-one mapping from ℝn to ℝm, then it must be the case that m >= n.

Here's why:

1. A one-to-one mapping means that each element in the domain has a unique image in the codomain. In other words, no two distinct points in the domain can have the same image.
2. If n > m, then there are more elements in the domain than in the codomain. This means that it is not possible to have a unique image for each element in the domain, and the mapping cannot be one-to-one. Therefore, m must be greater than or equal to n.
3. For example, let's consider a mapping from ℝ3 to ℝ2. The domain has three dimensions (x, y, z), while the codomain has only two dimensions (u, v). There is no way to differentiate between all the different points in the domain using only two dimensions, so the mapping cannot be one-to-one.