Calculating how many elements are in the product of Cartesian multiplication Let $A = \{1,3\}, B = \{1, 2\}, C = \{1, 2,3\}$. 
How many elements are there in the set
$\{(x,y,z) \in A \times B \times C  | x + y = z \} $ ? 
Two things I'm not familiar with here, 
First, how do I do Cartesian multiplication between 3 sets?
And I'm having trouble figuring how the $x+y=z$ has to do with the number of elements?
Can you please show me how to solve this? 
thanks :)

The Cartesian product of the three sets $A$, $B$, and $C$ is just the triplet $(a,b,c)$ such that $a∈ A$, $b∈B$, and $c∈C$. We wish to find any and all $(a,b,c)$ that satisfies the property $a+b=c$, where $a∈ A$, $b∈B$, and $c∈C$.
So $1+1=2$, and $1∈A$, $1∈B$, and $2∈C$, so we found one element, namely $(1,1,2)$.
$1+2 = 3$, and $1∈A$, $2∈B$, and $3∈C$, so we found another element, namely $(1,2,3)$.
Notice that $(1,1,3)$ is in $A×B×C$, but $1+1\neq3$, so $(1,1,3)$ is not an element.
Since our sets are small, we can repeat this process and then count how many elements you found that satisfy that property.
Edit: looks like that's all of them.
