In

_{1} and criteria are comparable, while the T_{1} and

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, two elements ''x'' and ''y'' of a set ''P'' are said to be comparable with respect to a binary relation
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Mathematics
* Binary number
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≤ if at least one of ''x'' ≤ ''y'' or ''y'' ≤ ''x'' is true. They are called incomparable if they are not comparable.
Rigorous definition

Abinary relation
Binary may refer to:
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Mathematics
* Binary number
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on a set $P$ is by definition any subset $R$ of $P\; \backslash times\; P.$ Given $x,\; y\; \backslash in\; P,$ $x\; R\; y$ is written if and only if $(x,\; y)\; \backslash in\; R,$ in which case $x$ is said to be to $y$ by $R.$
An element $x\; \backslash in\; P$ is said to be , or (), to an element $y\; \backslash in\; P$ if $x\; R\; y$ or $y\; R\; x.$
Often, a symbol indicating comparison, such as $\backslash ,<\backslash ,$ (or $\backslash ,\backslash leq\backslash ,,$ $\backslash ,>,\backslash ,$ $\backslash geq,$ and many others) is used instead of $R,$ in which case $x\; <\; y$ is written in place of $x\; R\; y,$ which is why the term "comparable" is used.
Comparability with respect to $R$ induces a canonical binary relation on $P$; specifically, the induced by $R$ is defined to be the set of all pairs $(x,\; y)\; \backslash in\; P\; \backslash times\; P$ such that $x$ is comparable to $y$; that is, such that at least one of $x\; R\; y$ and $y\; R\; x$ is true.
Similarly, the on $P$ induced by $R$ is defined to be the set of all pairs $(x,\; y)\; \backslash in\; P\; \backslash times\; P$ such that $x$ is incomparable to $y;$ that is, such that neither $x\; R\; y$ nor $y\; R\; x$ is true.
If the symbol $\backslash ,<\backslash ,$ is used in place of $\backslash ,\backslash leq\backslash ,$ then comparability with respect to $\backslash ,<\backslash ,$ is sometimes denoted by the symbol $\backslash overset$, and incomparability by the symbol $\backslash cancel\backslash !$.
Thus, for any two elements $x$ and $y$ of a partially ordered set, exactly one of $x\backslash \; \backslash overset\backslash \; y$ and $x\; \backslash cancely$ is true.
Example

Atotally ordered
In mathematics
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set is a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable.
Properties

Both of the relations and aresymmetric
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

, that is $x$ is comparable to $y$ if and only if $y$ is comparable to $x,$ and likewise for incomparability.
Comparability graphs

The comparability graph of a partially ordered set $P$ has as vertices the elements of $P$ and has as edges precisely those pairs $\backslash $ of elements for which $x\backslash \; \backslash overset\backslash \; y$..Classification

When classifying mathematical objects (e.g.,topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

s), two are said to be comparable when the objects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they are comparable under the partial order ⊂. For example, the Tsobriety
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Sobriety is the condition of not having any measurable levels or effects from alcohol or drugs. Sobriety is also considered ...

criteria are not.
See also

* , a partial ordering in which incomparability is atransitive relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

References

{{reflist Binary relations Order theory