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:
\(\ds V \times W\) | \(=\) | \(\ds \set {\tuple {v_1, w_1}, \tuple {v_1, w_2}, \tuple {v_1, w_3}, \tuple {v_2, w_1}, \tuple {v_2, w_2}, \tuple {v_2, w_3} }\) | ||||||||||||
\(\ds V \times V\) | \(=\) | \(\ds \set {\tuple {v_1, v_1}, \tuple {v_1, v_2}, \tuple {v_2, v_1}, \tuple {v_2, v_2} }\) |
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:
- $A \times B = \set {\tuple {1, 3}, \tuple {1, 4}, \tuple {2, 3}, \tuple {2, 4} }$