Cardinality of Cartesian Product where (a,b) are elements of A x B I'm having issues with a question regarding the cardinality of a cartesian product.
Question:
Let $A=\{0, 1, 2, 3, 4, 5, 7\}$ and $B=\{0, 2, 4, −1, 12\}.$
How many elements are in $\{(a, b) ∈ A × B \; |  \; a < 7 \text{ and } b < 4\}$?
I want this to be $6 \cdot 3 = 18$. but that seems to be wrong.
 A: You may begin by omitting all $a \in A$ such that $a\ge 7$ and all $b \in B$ such that $b \ge 4$ because we know ordered pairs with said elements will not be included in the Cartesian product we're looking for. So, after omission, we have $A = \{0,1,2,3,4,5\}$ and $B=\{0,2,-1\}$. Now, the cardinality of the Cartesian product between sets $A$ and $B$, denoted as $|A$ x $B|$, is given by $|A$ x $B|$ = $|A| \cdot |B| = 6 \cdot 3 = 18$. So you are correct! If your book says differently, then I highly suspect your book contains a mistake.
A: Why does it seem wrong?
$\{(a,b)\in A\times B| a<7$ and $b< 4\}=$
$\{(a,b)| a\in \{k\in A|k < 7\}$ and $b\in \{j\in B|j < 4\}=$
$\{(a,b)| a \in \{0,1,2,3,4,5\}$ and $b \in \{-1,0,2\}\}=$
$\{0,1,2,3,4,5\} \times  \{-1,0,2\}$
And if $C, D$ are finite then $|C\times D| = |C|\cdot |D|$.
And as $|\{0,1,2,3,4\}|= 6$ and $|\{-1,0,2\}| = 3$ we have
$|\{0,1,2,3,4,5\} \times  \{-1,0,2\}= 6\cdot 3 = 18$
If you want we can even list them all:
$(0,-1),(0,0), (0,2),(1,-1),(1,0),(1,2), (2,-1),(2,0), (2,2),(3,-1),(3,0),(3,2),(4,-1),(4,0), (4,2),(5,-1),(5,0),(5,2)$.
.... I .... really have no idea why you think this seems wrong.  There are very few things in my life right now that feel as right as this feels.
A: If I've understood your question correctly, you'll begin by taking the appropriate subsets of A and B. 
Since A = {0,1,2,3,4,5,7} and B = {0,2,4,−1,12}, and you're restricting your subsets to a < 7, and b < 4, this leaves the sets C = {0,1,2,3,4,5} and D = {0,2,-1}. 
If you allow for repetition, which your question seems to assume, then there are 6 ways to select the first element, and 3 ways to select the second, implying a total of 18 ways to construct ordered pairs on the subsets.
Note that your textbook might be looking for un-ordered subsets of size 2, which is a different question.
