In cartesian product, is the order of factors important?

Question

For example- $\{1,3\} \times \{1,2\}$. Here $4$ elements will be formed. So if i write $\{1,2\}$ before $\{1,3\}$, is there any difference?

Answer 1

Yes, the elements of a Cartesian product are ordered tuples.

The Cartesian product is not commutative.   The order of its operation is significant.

$$\rm \forall A\neq\emptyset~,\forall B\neq\emptyset\qquad A\neq B~\to~ A\times B\neq B\times A$$

As per your example:

$\{1,3\}\times\{1,2\} = \{(1,1),(1,2),(3,1),(3,2)\}$

$\{1,2\}\times\{1,3\} = \{(1,1),(1,3),(2,1),(2,3)\}$

Since the order in the tuples is significant, these are not the same sets.

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If, however, you meant the order of the tuples within the product; the product is a set so that is not important.

$\{(1,1),(1,2),(3,1),(3,2)\}=\{(1,1),(3,1),(1,2),(3,2)\}$ et cetera.

Answer 2

In Cartesian product of $\{1,3\} \times \{1,2\}$ order matters. So you can't write it as $\{1,2\} \times \{1,3\}$. But in the resultant set you can place elements at any position.