(Solved): Let V=R be the vector space over R equipped with the usual vector addition and scalar multiplicatio ...

Let $V=\mathbb{R}$ be the vector space over $\mathbb{R}$ equipped with the usual vector addition and scalar multiplication operations on $\mathbb{R}$. Determine if the following subsets of $V$ are subspaces. The set of positive real numbers ${\mathbb{R}}_{>0}=\{x?\mathbb{R}:x>0\}$ equipped with the usual vector addition and scalar multiplication operations. The set of rational numbers $\mathbb{Q}=\left\{\frac{x}{y}:x,y?\mathbb{Z},y?=0\right\}$ equipped with the usual vector addition and scalar multiplication operations. The set of positive real numbers ${\mathbb{R}}_{>0}=\{x?\mathbb{R}:x>0\}$ equipped with vector addition $x+y=xy$ and scalar multiplication $cx=x^c$. The set of integers $\mathbb{Z}=\{\dots ,?2,?1,0,1,2,\dots \}$ equipped with the usual vector addition and scalar multiplication operations. The set of real numbers $\mathbb{R}$ equipped with the usual vector addition and scalar multiplication operations. The set of complex numbers $\mathbb{C}=\{x+iy:x,y?\mathbb{R}\}$ equipped with the usual vector addition and scalar multiplication operations.