Relations and Functions - Cartesian Product of Sets

Ordered Pairs: An ordered pair consists of two objects or elements in a fixed order, typically written as $(a, b)$. In this pair, $a$ is called the first element and $b$ is the second element. Visually, this can be represented as a specific point on a 2D Cartesian plane, where switching the order to $(b, a)$ would result in a different point unless $a = b$.

Equality of Ordered Pairs: Two ordered pairs $(a_1, b_1)$ and $(a_2, b_2)$ are equal if and only if their corresponding first elements are equal and their corresponding second elements are equal, i.e., $a_1 = a_2$ and $b_1 = b_2$.

Cartesian Product of Two Sets: For two non-empty sets $A$ and $B$, the Cartesian product $A \\times B$ is the set of all ordered pairs $(a, b)$ such that $a \\in A$ and $b \\in B$. Conceptually, if you list elements of $A$ along a horizontal axis and elements of $B$ along a vertical axis, the Cartesian product represents every intersection point in the resulting grid.

Cardinality of Cartesian Product: If set $A$ has $p$ elements and set $B$ has $q$ elements, then the number of elements in the Cartesian product $A \\times B$ is the product of the number of elements in each set. This is expressed as $n(A \\times B) = n(A) \\times n(B)$.

Cartesian Product with Empty Set: If either set $A$ or set $B$ is an empty set (null set $\\phi$), then the Cartesian product $A \\times B$ is also an empty set, because no ordered pairs can be formed. Symbolically, $A \\times \\phi = \\phi$.

Cartesian Product of Three Sets: The product $A \\times B \\times C$ results in a set of ordered triplets $(a, b, c)$ where $a \\in A, b \\in B,$ and $c \\in C$. Visually, this extends the 2D grid concept into 3D space, where each triplet represents a unique point $(x, y, z)$ in a three-dimensional coordinate system.

Graphical Representation (Arrow Diagrams): A Cartesian product can be visualized using an arrow diagram where two closed loops represent sets $A$ and $B$. Every element in the first loop has an arrow pointing to every single element in the second loop, representing all possible pairings.