Cartesian Product/Examples

Examples of Cartesian Products

Product of Arbitrary Sets: 1

Let $S = \set {1, 2, 3}$.

Let $T = \set {a, b}$.

Then:

\(\ds S \times T\)

\(=\)

\(\ds \set {\tuple {1, a}, \tuple {1, b}, \tuple {2, a}, \tuple {2, b}, \tuple {3, a}, \tuple {3, b} }\)

\(\ds T \times S\)

\(=\)

\(\ds \set {\tuple {a, 1}, \tuple {a, 2}, \tuple {a, 3}, \tuple {b, 1}, \tuple {b, 2}, \tuple {b, 3} }\)

Product of Arbitrary Sets: 2

Let $V = \set {v_1, v_2}$.

Let $W = \set {w_1, w_2, w_3}$.

Then:

\(\ds V \times W\)

\(=\)

\(\ds \set {\tuple {v_1, w_1}, \tuple {v_1, w_2}, \tuple {v_1, w_3}, \tuple {v_2, w_1}, \tuple {v_2, w_2}, \tuple {v_2, w_3} }\)

\(\ds V \times V\)

\(=\)

\(\ds \set {\tuple {v_1, v_1}, \tuple {v_1, v_2}, \tuple {v_2, v_1}, \tuple {v_2, v_2} }\)

Arbitrary Example $1$

Let:

\(\ds A\)

\(=\)

\(\ds \set {1, 2}\)

\(\ds B\)

\(=\)

\(\ds \set {2, 3}\)

Then:

$A \times B = \set {\tuple {1, 2}, \tuple {1, 3}, \tuple {2, 2}, \tuple {2, 3} }$

Arbitrary Example $2$

Let:

\(\ds A\)

\(=\)

\(\ds \set {1, 2}\)

\(\ds B\)

\(=\)

\(\ds \set {3, 4}\)

Then:

$A \times B = \set {\tuple {1, 3}, \tuple {1, 4}, \tuple {2, 3}, \tuple {2, 4} }$