How to find number of maps from set $A$ to set $B$

Question

I was trying to solve the following question:

• For two sets $A$ and $B$ of cardinalities $3$ and $2$, how many maps from $A$ to $B$ exist?

• the correct answer: $8$

But I don't see why it is $8$. Please anyone explain. Also here, what does map means?

Answer 1

First of all, map is another way of saying function, a relation between A and B such that each element in A is connected to only one element in B.

Select an element from the set A, let's call it a. The map must connect a to one of the elements of B, so there are 2 choices for a. Since a is arbitrary, there are 2 choices for each element.

Therefore, first we select an element from B for a, then we select an element from B for the second element of A, and then the third one. As a result, there are

2 x 2 x 2 = $2^3$

different maps from A to B.

Answer 2

The set $A$ has $3$ elements. If $f$ is a map from $A$ to $B$, then, since $B$ has two elements, for each element $a\in A$, you have two possibilities for $f(a)$. So, you have $2\times2\times2=8$ ways of defining $f$.