# Functions and sets

#### Aleks855

Hey, I'm struggling a bit with this problem.

"Describe all functions $$\displaystyle f\ : \ A \ \right \ B$$ where $$\displaystyle A=\{1, 2, 3\}$$ and $$\displaystyle B=\{a, b\}$$".

After which we're going to debate whether or not each function is injective and/or surjective. But my real problem is that I don't know how to describe all the functions.

#### HallsofIvy

Math Team
A function, from set A to set B, is, pretty much by definition, a collection of "ordered pairs" in which the first member of every pair is in set A and the second member in set B. So one way of writing a function from {1, 2, 3} to {a,b} is to write three pairs {(1, ), (2, ), (3, )}. Then choose either a or b for each pair. That means you have three choices to make between a and b so there are $$\displaystyle 2^3= 8$$ possible functions here.
One of them is {(1, a), (2, a), (3, a)} and another is {(1, b), (2, b), (3, b)}. Can you find the other 6?

#### Aleks855

Aye, got it now, thanks =)

The rest would be the surjective functions, I suppose