If ${(a,b)}$ and ${(c,d)}$ are points in $ℝ^2$ then let $S$ be the set of point on the line segment that joins ${(a,b)}$ and ${(c,d)}$. Show $|S|= |ℝ|$
I can see this is similar to how the tangent function maps from the inverval ${(-\frac{\pi}2},\frac{\pi}2)$ to the set of real numbers but i'm not sure how to show that this is general for any line segment?
Added:
Could we write this line segment as an interval: $[(a,b),(c,d)]=(a+t(c-a),b+t(d-b)$ where $t \in [0,1]$ and then use the theorem that says $|[e,f]| = |[0,1]|$ where we let $e=a+t(c-a)$ and $f=b+t(d-b)$ and then from this use the theorem that states the cardinality of the set of all real numbers is the same as the cardinality of $[0,1]$? This almost seems too simple because I just strung together some theorems?
Hint: Write down a precise parametrization (bijection) of your segment by an interval in $\mathbb{R}$