Set of all versus some

Question

I was looking at the definition of a cartesian cross of two sets on Wikipedia earlier. $A \times B = \{(a, b) | a \in A \text{ and } b \in B\}$. However, I'm not sure how this defines a proper cross. Shouldn't there be some notion that these are for all a and b? Otherwise how could we define a set that had only some of the members of A and B? A bit confused here. The reason I ask is because in my class, my teacher defined the set E of a graph with set V vertices as: $E = \{(u, v) | u, v \in V\}$. But this was supposed to describe all directed graphs not just fully connected ones.

Answer 1

No.

The definition of $A\times B$, in words, can be written as

The set of all pairs $(a,b)$ such that $a\in A$ and $b\in B$.

Here "all" comes before the $(a,b)$. This is standard in set-builder notation. For example, $X=\{(1,2), (1,3)\}$ can be described by

The set of all pairs $(a,b)$ such that $a\in \{1\}$ and $b\in\{2,3\}$.

This completely defines $X$ and so

$$X=\{1\}\times \{2,3\}.$$

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We can describe a set $Y$ containing only some pairs of elements of $A$ and $B$ as

$$Y\subset A\times B.$$

Can you see why?

Answer 2

The set $\{(a,b) : a \in A \text{ and } b \in B\}$ is interpreted as "the set of all $(a,b)$ such that $a \in A$ and $b \in B$". It is implicit in the definition that this is for all $a \in A$ and $b \in B$.