00:01
In this problem, we need to prove that the open interval ab is uncountable.
00:05
Now, for that, first of all, we'll find a bijection from the open interval 0 -1 to ab.
00:14
So let us consider this function f from 0 -1 to the open interval ab, defined as f of x is equal to b minus e x plus a.
00:31
Now, we need to prove that it is a bijection.
00:35
Now, first of all, note that this is a linear function.
00:39
We have x to the power 1, and it is in the form fx equals to mx plus c.
00:49
So it is a linear function.
00:51
And all linear functions are 1 -1.
00:54
Because of that, the function f -of -x is 1 -1.
00:58
Next, we need to show that it is on to.
01:01
Now for that, let us consider any element y in the codomain ab and find whether it has a pre -image in the domain 0 -1.
01:18
So let y belong to ab says that y is equal to f of x.
01:24
So that will mean that y is equal to b minus a times x plus a...