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The Cartesian product of two sets A and B, denoted $A \times B$, is the set of all possible ordered pairs (x, y) where the first element x belongs to set A and the second element y belongs to set B.

Conditions: A and B are arbitrary sets.

The Cartesian product of two sets A and B, denoted $A \times B$, is the set of all possible ordered pairs (x, y) where the first element x belongs to set A and the second element y belongs to set B.

Conditions: A and B are arbitrary sets.

No, a relation does not have to contain every possible ordered pair. By definition, a relation is a subset of the Cartesian product $A \times B$, meaning it can include only some of the ordered pairs.

Conditions: A and B are sets.; R is a relation between A and B.

No, a relation does not have to contain every possible ordered pair. By definition, a relation is a subset of the Cartesian product $A \times B$, meaning it can include only some of the ordered pairs.

Conditions: A and B are sets.; R is a relation between A and B.

In the definition of a relation, capital letters A and B denote the sets themselves, while lowercase letters a and b denote individual elements belonging to those sets. The ordered pair (a, b) represents a specific element of the Cartesian product $A \times B$, where a is an element of A and b is an element of B.

Conditions: The context is the formal definition of a relation R between sets A and B.; The notation (a, b) ∈ $A \times B$ is used.

In the definition of a relation, capital letters A and B denote the sets themselves, while lowercase letters a and b denote individual elements belonging to those sets. The ordered pair (a, b) represents a specific element of the Cartesian product $A \times B$, where a is an element of A and b is an element of B.

Conditions: The context is the formal definition of a relation R between sets A and B.; The notation (a, b) ∈ $A \times B$ is used.

Yes, elements of the sets can be left out of a relation. A relation is a subset of the Cartesian product, so it does not need to include every possible ordered pair.

Conditions: A relation R is defined between sets A and B.; R is a subset of $A \times B$.

Yes, elements of the sets can be left out of a relation. A relation is a subset of the Cartesian product, so it does not need to include every possible ordered pair.

Conditions: A relation R is defined between sets A and B.; R is a subset of $A \times B$.

A relation R between two sets A and B is formally defined as a subset of the Cartesian product $A \times B$. This means that R is a collection of ordered pairs, where the first element of each pair belongs to set A and the second element belongs to set B.

Conditions: A and B are sets.; R is a binary relation between A and B.

A relation R between two sets A and B is formally defined as a subset of the Cartesian product $A \times B$. This means that R is a collection of ordered pairs, where the first element of each pair belongs to set A and the second element belongs to set B.

Conditions: A and B are sets.; R is a binary relation between A and B.

The Cartesian product distributes over both union and intersection from both the left and the right. Specifically, (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$), (A ∩ B) × $C = (A \times C)$ ∩ ($B \times C$), A × (B ∪ C) = ($A \times B$) ∪ ($A \times C$), and A × (B ∩ C) = ($A \times B$) ∩ ($A \times C$).

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

The Cartesian product distributes over both union and intersection from both the left and the right. Specifically, (A ∪ B) × $C = (A \times C)$ ∪ ($B \times C$), (A ∩ B) × $C = (A \times C)$ ∩ ($B \times C$), A × (B ∪ C) = ($A \times B$) ∪ ($A \times C$), and A × (B ∩ C) = ($A \times B$) ∩ ($A \times C$).

Conditions: A, B, and C 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.

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 $a \in A$.