## Relations

A **relation** from a set \( \sA \) to a set \( \sB \) is a subset of \( \sA \times \sB \), where \( \times \) denotes the Cartesian product.

#### Example

Consider the sets \( \sA = \lbrace 1, 2, 3 \rbrace \) and \( \sB = \lbrace x, y, z \rbrace \).

$$ R_1 = \lbrace (1,x), (2, y), (1, z) \rbrace $$
$$ R_2 = \lbrace (x,1), (y,2), (z,3) \rbrace $$
$$ R_3 = \lbrace (1,1), (2, y), (3, z) \rbrace $$

Among these, \( R_1 \) is a relation from \( \sA \) to \( \sB \) and \( R_2 \) is a relation from \( \sB \) to \( \sA \).
The set \( R_3 \) is not a relation between \( \sA \) and \( \sB \) because it contains the ordered pair \( (1,1) \).