Recall that for $a,b$ an element of the set of real numbers and $a<b$, the closed interval $[a,b]$ in real numbers is defined by $[a,b]=\{x$ is an element of reals|$a\leq x\leq b\}$. Show that the given intervals have the same cardinality by giving a formula for a one to one function $f$ mapping the first interval onto the second:
$[0,1] \to [0,2]$
So I understand that in order to show a function is one to one you need to show that the two sets have a one to one correspondence. How would you show this in a formal proof? I can only think of lining up the numbers in each set…
Best Answer
In the particular case of two closed intervals a bijection $[a,b]\to[c,d]$ easy to find is$$x\mapsto c+\frac{x-a}{b-a}(d-c)$$which "rescale" $[a,b]$ onto $[c,d]$.
A bijective function is a one-to-one correspondence, since it is both an injective, i.e. one-to-one, function and a surjective one.