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The intersection of two Cartesian products ($A \times B$) and ($C \times D$) is equal to the Cartesian product of the intersections of their respective component sets, i.e., (A ∩ C) × (B ∩ D).

Conditions: A, B, C, and D are arbitrary sets.

The intersection of two Cartesian products ($A \times B$) and ($C \times D$) is equal to the Cartesian product of the intersections of their respective component sets, i.e., (A ∩ C) × (B ∩ D).

Conditions: A, B, C, and D are arbitrary sets.

A group action of a group $G$ on a set $A$ is a map from $G \times A$ to $A$, written as $g \cdot a$, that satisfies two properties: compatibility with the group operation ($g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$) and the identity property ($1 \cdot a = a$).

Conditions: $G$ is a group.; $A$ is a set.; The action is a map from $G \times A$ to $A$.; Property (a) holds for all $g_1, g_2 \in G$ and $a \in A$.; Property (b) holds for all $a \in A$.

A group action of a group $G$ on a set $A$ is a map from $G \times A$ to $A$, written as $g \cdot a$, that satisfies two properties: compatibility with the group operation ($g_1 \cdot (g_2 \cdot a) = (g_1 g_2) \cdot a$) and the identity property ($1 \cdot a = a$).

Conditions: $G$ is a group.; $A$ is a set.; The action is a map from $G \times A$ to $A$.; Property (a) holds for all $g_1, g_2 \in G$ and $a \in A$.; Property (b) holds for all $a \in A$.