Can someone clarify the definition for two Latin squares to be orthogonal for me please?

Question

I have been given this definition for two Latin Sqaures to be orthogonal. Let $L, M$ be two $n\times n$ Latin squares with entries taken from the sets $X = \{x_1, x_2,\dots, x_n\}$,$\ \ Y = \{y_1, y_2,\dots, y_n\}$ respectively and $(n > 1)$. Then $L, M$ are said to be orthogonal if

$$\{ (L_{ij} , M_{ij} ) \mid| 1 ≤ i, j ≤ n\} = X × Y.$$

From the examples I've seen, $X\times Y$ does not seem to mean matrix multiplication. Hence can someone clarify this definition for me please?

Here may be help you. – Hoseyn Heydari Dec 22 '14 at 13:53 — Hoseyn Heydari, Dec 22 '14 at 13:53

score 2 · Accepted Answer · answered Dec 22 '14 at 13:53

Here $\times$ is the Cartesian product, the set of all pairs $(x,y)$ where $x \in X$ and $y \in Y$. Two latin squares are orthogonal if when superimposed on each other, the pairs of numbers appearing in each cell never repeat, or equivalently for each $x \in X$ and $y \in Y$ there is a cell which contains both.

Can someone clarify the definition for two Latin squares to be orthogonal for me please?

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