Ordered Pairs, Cartesian Product and Plane

 

Ordered Pairs

an ordering of objects such that order of the objects matter

$(1, 2) \not = (2, 1)$

Ordered n-Tuple

$(x_1, x_2, x_3, \cdots, x_n) = (y_1, y_2, y_3, \cdots, y_n)$ if $x_1 = y_1$

$(a, b)$ : ordered 2-Tuple

$(a, b, c)$ : ordered 3-Tuple

 

 

Cartesian Product

Given sets $A$ and $B$, the Cartesian Product of $A$ and $B$, denoted $A \times B$ and read "$A$ cross $B$", is the set of all ordered pairs $(a, b)$, where $b$ is in $A$ and $b$ is in $B$

$A \times B = \{ (a, b)\ |\ a \in A\ and\ b \in B \}$

 

$e.g.\quad A = (a, b), B = (1, 2),\ then\ A\times B = \{(a, 1), (b, 1), (a, 2), (b, 2)\}$

 

make sure that $A \times B \not = B \times A$

because it has orders

 

 

$A \times B \times C= \{ (a, b, c)\ |\ a \in A,\ \ b \in B\ and\ c \in C \}$

 

$e.g.\quad A = (a, b), B = (1, 2),\ C = (x, y)$

$then\\ A\times B \times C = \{(a, 1, x), (a, 1, y), (b, 1, x), (b, 1, y), (a, 2, x), (a, 2, y), (b, 2, x), (b, 2, y)\}$

 

make sure that $A\times B \times C \not = (A\times B) \times C$

$(A\times B) \times C = \{((a, 1), x), ((a, 1), y), ((b, 1), x), ((b, 1), y), ((a, 2), x), ((a, 2), y), ((b, 2), x), ((b, 2), y)\}$

nested tuples are made

 

 

Cartesian Plane

Cartesian Plane (=Coordinate Plane) (named after Rene Decartes)

$R \times R$ → 2D Cartesian Plane

$R\times R\times R$ → 3D Cartesian Plane