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indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation be transitive: for all if and then |

In mathematics, a relation on a set is called **connected** or **complete** or **total** if it relates (or "compares") all *distinct* pairs of elements of the set in one direction or the other while it is called **strongly connected** if it relates *all* pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all there is a so that (see serial relation).

Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain).

A relation on a set is called ** connected** when for all

or, equivalently, when for all

A relation with the property that for all

is called

The main use of the notion of connected relation is in the context of orders, where it is used to define total, or linear, orders. In this context, the property is often not specifically named. Rather, total orders are defined as partial orders in which any two elements are comparable.^{[4]}^{[5]}
Thus, *total* is used more generally for relations that are connected or strongly connected.^{[6]} However, this notion of "total relation" must be distinguished from the property of being serial, which is also called total. Similarly, connected relations are sometimes called *complete*,^{[7]} although this, too, can lead to confusion: The universal relation is also called complete,^{[8]} and "complete" has several other meanings in order theory.
Connected relations are also called *connex*^{[9]}^{[10]} or said to satisfy *trichotomy*^{[11]} (although the more common definition of trichotomy is stronger in that *exactly one* of the three options must hold).

When the relations considered are not orders, being connected and being strongly connected are importantly different properties. Sources which define both then use pairs of terms such as *weakly connected* and *connected*,^{[12]} *complete* and *strongly complete*,^{[13]} *total* and *complete*,^{[6]} *semiconnex* and *connex*,^{[14]} or *connex* and *strictly connex*,^{[15]} respectively, as alternative names for the notions of connected and strongly connected as defined above.

Let be a homogeneous relation. The following are equivalent:^{[14]}

- is strongly connected;
- ;
- ;
- is asymmetric,

where is the universal relation and is the converse relation of

The following are equivalent:^{[14]}

- is connected;
- ;
- ;
- is antisymmetric,

where is the complementary relation of the identity relation and is the converse relation of

Introducing progressions, Russell invoked the axiom of connection:

Whenever a series is originally given by a transitive asymmetrical relation, we can express connection by the condition that any two terms of our series are to have the generating relation.

— Bertrand Russell,The Principles of Mathematics, page 239

- The
*edge*relation^{[note 1]}of a tournament graph is always a connected relation on the set of 's vertices. - If a strongly connected relation is symmetric, it is the universal relation.
- A relation is strongly connected if, and only if, it is connected and reflexive.
^{[proof 1]} - A connected relation on a set cannot be antitransitive, provided has at least 4 elements.
^{[16]}On a 3-element set for example, the relation has both properties. - If is a connected relation on then all, or all but one, elements of are in the range of
^{[proof 2]}Similarly, all, or all but one, elements of are in the domain of