Math interval notations: Interval raised to a power?

Question

I saw this equation in a paper:

$v = (v_{l,0}, v_{r,0}, v_{l,1}, v_{r,1}, v_{l,2}, v_{r,2}, v_{l,3}, v_{r,3}) ∈ [−1, 1]^8 $

What does this interval notation mean?- $[−1, 1]^8$

Answer 1

It is exponentiation in the context of the Cartesian product of sets. For example: \begin{align} [-1, 1]^3 &= [-1, 1] \times [-1, 1] \times [-1, 1] \\ &= \{ (x_1, x_2, x_3) \mid x_1, x_2, x_3 \in [-1, 1] \} \end{align}

Answer 2

$$[-1,1]^8=\left\{ (a,b,c,d,e,f,g,h) : a\in[-1,1] \text{ and }b\in[-1,1]\text{ and } \cdots \text{ and } h\in[-1,1]\right\}$$

It’s the interval version of the Cartesian product.

Formally, one should use $\wedge$ instead of “$\text{ and }$.”

Answer 3

[-1, 1] represents a subspace of 1-Dimensional space ( R ) where x₁ is constrained to lie in [-1, 1].

Similarly, [-1, 1]^⁸ represents subspace of 8-Dimensional space ( R^⁸ ), where each of the x_i's is bound to lie in [-1, 1].

In other words, a 8-tuple ( x₁, x₂, ... x₈ ) will lie in [-1, 1]^⁸ if each of the x_i's lies in [-1, 1].

Answer 4

The previous answers are wrong. Let w be the interval [-2,3]. Then the meaning of w^2 is this: Let L= the minimum value of x^2 for x in w. let H = the maximum value of x^2 for x in w. Then the interval result for w^2 is {L,H]. In this case the smallest value is 0, the largest is 9, so the answer is [0,9]. By contrast, [-2,3] ^2 is the interval [-6,9]. RJF