Let \(A\) be a set. A relation \(R\) on \(A\) is called a partial ordering if it satisfies the following:

1. \((a,a) \in R\) for every \(a \in A.\)
2. If \((a,b) \in R\) and \((b,a) \in R\) then \(a = b.\)
3. Transitivity: If \((a,b) \in R\) and \((b,c) \in R\) then \((a,c) \in R.\)

If, in addition, for every \(a, b \in A\) at least one of \((a,b) \in R\) or \((b,a) \in R\) then \(R\) is called an ordering.

Example: Let \(S = \{1,2,3\}\) and let \(A = \mathcal{P}(S)\) be the power set of \(S.\) The operation \(R = \subset\) is a partial ordering.

In this example, not every pair of elements can be compared. For example, \(\{1,2\}\) and \(\{1,3\}\) since neither set is a subset of the other.

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The set \(A = \{1,2,3,4\}\) with \(R = \leq\) is an ordering. Any two elements of \(A\) can be compared.