Past exam of the logic and set theory course of the University of Oxford / 2009 / B / 5 / v / Solution

The common amplitude $a>0$ is uniquely determined by any entry's absolute value. After choosing it, each entry independently has sign $+1$ or $-1$. Thus the correspondenceis a bijection from $(0,\infty )\times \{-1,1{\}}^{\mathbb{N}}$ onto the set. The positive reals have cardinal $\mathfrak{c}$, and binary sign sequences have cardinal $2^{{\aleph }_0}=\mathfrak{c}$. HenceHere “independently” refers to unconstrained deterministic choices of coordinates, not probabilistic independence; no probability distribution is being specified.