The cross product of two sets Just reading about the cross product. It says here:
$$A\times B= \{(x,y)|x\in A, y\in B\}$$
Does this mean that the result of $A\times B$ is a new set of ordered pairs? If so, what happens if both A and B share a common member? What would the result be, since an ordered pair cannot include duplicates?



Edit: Quoting the free textbook I'm looking at online,


  If we are given that A is a set and no other information about A, then there is no
  ordering to the elements of A. Thus, we cannot speak of “the second element of the set A”
  unless we have specified an ordering of the elements of A. If we wish to regard A as ordered in some way, then we specify this fact explicitly: “The elements of A are ordered a, b, c,” or “A = (a, b, c).” The latter notation replaces the braces with parentheses and designates that A is ordered, left to right, as indicated. We call this an ordered set. An ordered set is also called a linear order. Various other names are also used: list, vector, string, word — all with no repeated elements. Of course, you’ve seen repeated elements in vectors, for example the point in the plane at the coordinates (1,1). That’s fine, it’s just not an ordered set. 
  
  Why is it okay to specify a set S = {a, b, c, a} where the element a has been repeated,
  but it is not okay to have repeated elements in an ordering of S?   When we say S  =
  {a, b, c, a}, we know that S contains just the three elements a, b and c.  If we were to talk
  about the ordered set (a, b, c, a) it would not make sense because it would say that the
  element a is in two places at once: the first position and the last position.
Yes. $A \times B$ is a set of ordered pairs of elements from $A$ and $B$. If $x \in A$ and $x \in B$, then the ordered pair generated will be $(x,x)$ which is a valid ordered pair and is an element of $A \times B$.

For instance, if $A = \{0,1\}$ and $B = \{1,2\}$, then $A \times B = \{(0,1),(0,2),(1,1),(1,2)\}$. Hope this clarifies.
No,This is no the real multiplication of sets.The real multiplication of sets is:
f(x)   multiplicity of x
A*B={x+y:f(x)f(y)|x:f(x)∈A,y:f(y)∈B}
when f(x)=f(y)=1
AB={x+y|x∈A,y∈B}
why?because:
P({a,b})={0,{a},{b},{a,b}}
P({c,d})={0,{c},{d},{c,d}}
P({a,b,c,d})={0,{c},{d},{c,d},{a},{a,c},{a,d},{a,c,d},{b},{b,c},{b,d},{b,c,d},{a,b},{a,b,c},{a,b,d},{a,b,c,d}}
P(A+B)=P(A)*P(B)
People do not know the real multiplication of sets.