Cartesian Product

A Cartesian Product of two #202204281446 \(A\) and \(B\), is the set of all #202204281552 \((a, b)\) where \(a \in A\) and \(b \in B\). It is denoted as \(A \times B\) which read as “A cross B”. This could be expressed in 202204281700# as \(A \times B = \{(a, b) \in S|a \in A \ \text{and}\ b \in B\}\).

Note: The number of elements of set \(A \times B\) is the product of the number of elements of set \(A\) and \(B\). (\(n(A \times B) = n(A) \cdot n(B)\))

The Cartesian Product of two \(\mathbb{R}\) (from 202204281506#) could be displayed in 202204281721# as a coordinate.

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  • Relation

    Let \(A = \{0, 1, 2\}\) and \(B = \{1, 2, 3\}\), and let’s say that element \(a\) in \(A\) is related to element \(b\) in \(B\) if and only if \(a < b\). This relation could be expressed in a set as \(\{(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)\}\). The result of \(A \times B\) is \(\{(0, 1), (0, 2), (0, 3), (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)\}\). It is not hard to see that it is a 202204281535 of 202204281601# of \(A\) and \(B\) aka \(A \times B\).

  • Function

    Based on this definition, the unique element in \(B\) related to \(x\) by \(F\) is denoted \(F(x)\), read “F of x”. We can understand it as \(F(x)\) is the output or result of input \(x\) in the function \(F\). A function could be express in 202204281700 as \(F = \{(x, y) \in A \times B|y = F(x)\}\). (relation is a 202204281535 of 202204281601 of \(A\) and \(B\))

#math #)