Effect of using of brackets in Cartesian Product?

Question

If $A$ and $B$ are any sets, then is it that the cartesian product of $A \times A \times B$ does not equal the cartesian product of $A \times ( A \times B)$?

Answer 1

The definition of Cartesian Product goes as follows.

Definition: Let $A$ and $B$ be non-empty sets. The Cartesian Product of $A$ and $B$ is the set
$$A \times B = \{(a,b) \mid a \in A \wedge b \in B\},$$ where $(a,b)$ denotes an ordered pair.

To understand better what we are doing here with ordered pairs, let’s go back to the definition of ordered pair given by Kuratowski.

Definition: Let $a, b$ be any objects. We define the ordered pair of $a$ and $b$, denoted by $(a,b)$ to be the set
$$(a,b) = \{\{a\},\{a,b\}\}.$$

Now, let $A$, $B$ and $C$ be non-empty sets. Consider now the Cartesian Products $A \times (B \times C)$ and $(A \times B) \times C$.

If $a \in A$, $b \in B$ and $c \in C$, then $(a,(b,c))$ denotes an element of $A \times (B \times C)$ and $((a,b),c)$ denotes an element of $(A \times B) \times C$.

Strictly speaking, $(a,(b,c)) \neq ((a,b),c).$ You can check this by Kuratowski’s definition.

$$(a,(b,c)) = \{\{a\},\{a,(b,c)\}\} = \{\{a\},\{a,\{\{b\},\{b,c\}\}\}\}.$$

$$((a,b),c) = \{\{(a,b)\},\{(a,b),c\}\} = \{\{\{\{a\},\{a,b\}\}\},\{\{\{a\},\{a,b\}\},c\}\}.$$

Note that $\{a\} \in (a,(b,c))$ but $\{a\} \notin ((a,b),c)$ so they are different by definition of equality of sets. Which means that $A \times (B \times C) \neq (A \times B) \times C$.

Although, we can easily see that $A \times (B \times C) \cong (A \times B) \times C$, i.e., $A \times (B \times C)$ and $(A \times B) \times C$ are isomorphic.

Consider $f:A \times (B \times C) \to (A \times B) \times C$ defined by $f((a,(b,c))) = ((a,b),c)$ for all $(a,(b,c)) \in A \times (B \times C)$.

The fact that they are isomorphic allows us to treat them as “equal”, as being the same set. This proves that the Cartesian Product is Associative up to an isomorphism and so it makes sense to define $A \times B \times C$ to be equal to $A \times (B \times C)$ or $(A \times B) \times C$. And so we defined it as follows.

$$A \times B \times C = \{(a,b,c) \mid a \in A \wedge b \in B \wedge c \in C\}.$$

Finally, note that this definition is indeed unambiguous from what we have just seen.