Cardinality Inequality Proof
Prove that .
To prove this inequality I need to show there exists a 1-1 function . And that all functions cannot be ONTO.
I'm having trouble conceptualizing a 1-1 function that maps from . However to the best of my knowledge I need a function that takes values from the interval [0,1] as input and outputs a function from the set of functions . A simple function in that I'm thinking of is just a straight line, so where b acts like the y-intercept. So my idea for a 1-1 mapping from , would take inputs from [0,1] and map them to various parallel lines, with different values for b. So basically . So my function is mapping to various parallel lines, I think that insures that f is 1-1.
If my 1-1 function is indeed correct, now I need to prove all functions cannot be ONTO. And I can't think of a way to start making progress on this part.
Thanks in advance for the help.