Cartesian Product/Examples

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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:

\(\ds V \times W\) \(=\) \(\ds \set {\tuple {v_1, w_1}, \tuple {v_1, w_2}, \tuple {v_1, w_3}, \tuple {v_2, w_1}, \tuple {v_2, w_2}, \tuple {v_2, w_3} }\)
\(\ds V \times V\) \(=\) \(\ds \set {\tuple {v_1, v_1}, \tuple {v_1, v_2}, \tuple {v_2, v_1}, \tuple {v_2, v_2} }\)