When all the elements of a #202204281446 \(S\) is also an element in another set \(P\), we could say that \(S\) is a subset of \(P\) which is denoted as \(S \subseteq P\). From this property, we know that \(P\) is also a subset of itself, thus \(P \subseteq P\).
If at least one element of \(P\) is not in the subset \(S\), then \(S\) is a proper subset of \(P\), denoted as \(S \subset P\).
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 \(A \times B\).
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 \(B\))