1. ## Surjection/bijection problem

Hi, I've been reading up on injection, surjection and bijection, but I haven't been able to find any examples that can help me figure out the solutions for these two:

1. Let A = {1, 2, 3} and B = {a, b, c, d}. Find one surjection from B to A.

2. Is the following injection and surjection (bijection): f: (0,1) -> (1, e), f(x) = ex

Any help is much appreciated

2. ## Re: Surjection/bijection problem

The diagram should help with the first question ...

For the second question, sketch the graph of $y=e^x$ over the interval $x \in (0,1)$ and note the range.

3. ## Re: Surjection/bijection problem

"Surjection", also called "onto" (as in "f is a function from B onto A), simply means that, for every member of A, y, there is a member of B, x, such that f(x)= y. Given that B= {a, b, c, d} and A= {1, 2, 3}, all you have to do is write f(a)= _, f(b)= _, f(c)= _, and f(d)= _, filling in the blanks with members of A, 1, 2, or 3. There are many ways to do that (specifically [tex]4^3= 256 ways). Choose one!

For the second, with $f(x)= e^x$, to determine if it is an "injection", you need to answer: suppose f(x)= f(y) (that is that $e^x= e^y$. Is it necessarily true that x= y? To determine if it is a "bijection", suppose y is any number between 1 and e and that $e^x= y$. Can you solve for x? Is x between 0 and 1?