Chapter 4.2, Problem 20P

On ${ℝ}^{+}$, the set of positive real numbers, define the operations of addition, $\oplus$, and scalar multiplication, $\odot$, as follows:

$x\oplus y=xy$

$c\odot x=x^c$.

Note that the multiplication and exponentiation appearing on the right side of these formulas refer to the ordinary operations on real numbers. Determine whether ${ℝ}^{+}$, together with these algebraic operations, is a vector space.