# cardinality of sets

Is it true that $\mathbb{|R|} < \mathbb{|R}^2|$
No. $\mathbb{R}$ is equinumerous with (0,1); the bijection between these sets can be constructed using the tangent function on $(-\pi/2,\pi/2)$. Similarly, $\mathbb{R}^2$ is equinumerous with $(0,1)\times(0,1)$. Now, there is a bijection $(0,1)\to(0,1)\times(0,1)$. See section 4 in this PDF document.