CSCI 2824 Lecture 19: Properties of Relations

We will now restrict ourselves to relations R subseteq A times A for a set A. Such relations can be viewed as a graph.

Graphs and Relations

Relations R subseteq A times A with domain and co-domain as the same set A can be viewed as a graph.


A graph G consists of a set A of nodes and a relation R subseteq A times A of edges. Each edge in a graph a rightarrow b corresponds to a pair (a,b) in R.

It is a common convention to call the set of nodes V (rather than A) edge relation E subseteq V times V (rather than Rsubseteq A times A) if we are talking about a graph. But we will ignore this convention for now.

We will look at two examples of relations and their corresponding graphs.

Example # 1

Consider the relation R_1: { (a,a), (b,a), (c,a), (d,a), (d,b), (d,c), (d,d) } over the set of nodes A = { a,b,c,d,e }. Its graph is depicted below:


Note that the graph has a node e without any links. This is because ein A but there is no tuple involving e in the relation R that describes the edge set.

Next, note that the edges corresponding to (a,a) and (d,d) are called self-loops. The graph of the relation in this example has two self loops, one over a and the other over d.

Example # 2

Consider the relation R_2: { (1,1), (1,2), (2,1), (2,3), (3,2), (3,3), (2,2) } over the set of nodes B = { 1,2,3 }. Its graph is depicted below:


Note that the arrow from 1 to 2 corresponds to the tuple  (1,2) in R_2, whereas the reverse arrow from 2 to 1 corresponds to the tuple (2,1) in R_2.

Types of Relations

We first study three types of relations: reflexive, symmetric and transitive.

Reflexive Relation

A relation R is reflexive iff (a,a) in R for all a in A.

From the graph, we note that a relation is reflexive if all nodes in the graph have self-loops

The relation R_2 from example #2 above is reflexive whereas the relation R_1 from example #1 is not. R_1 is missing the self loops from (b,b), (c,c) and (e,e).

The next concept is that of a symmetric relation.

Symmetric Relation

A relation R is symmetric iff for all (a,b) in R, (b,a) in R.

From its graph, a relation is symmetric if for every “forward” arrow from a to b, there is also a reverse arrow from b to a.

The relation R_2 from example #2 above is symmetric whereas the relation R_1 from example #1 is not. R_1 has the edge (d,b) but not the reverse edge (b,d).

Finally, we will talk about transitive relations.

Transitive Relation

A relation R is transitive iff for all a,b,c, IF (a,b) in R and (b,c) in R THEN (a,c)in R.

In graph terms, if we start at some node a, and using an edge go from a to b and then from b to c using another edge, we should also be able to go from a to c directly using an edge.

The relation R_2 from example #2 above is not transitive. It is missing the edges (1,3) and (3,1), that would make it transitive. The relation R_1 from example #1 is transitive, on the other hand.

Putting all these together, a relation is an equivalence iff it is reflexive, symmetric and transitive.

We now consider the polar opposite of a reflexive relation, an irreflexive relation:

Irreflexive Relation

A relation R is irreflexive iff (a,a) notin R for all a in A.

For all a in A, we have (a,a) notin R .

While a reflexive relation has all the self-loops, an irreflexive one has no self-loops.

The relation R_1 in example # 1 is not irreflexive since it has self-loops (a,a) and (d,d)$. Removing these from the relations yields us an irreflexive relation


Take the set A = {1,2,3,4}. Give us examples of relations that are

  • Reflexive: {(1,1), (2,2), (3,3), (3,4), (4,4) }.

  • Irreflexive: {(1,3)}. Caution Irreflexive is not the logical negation of reflexive. It is stronger than that.

  • Symmetric: {(1,2), (2,1), (1,1), (1,4), (4,1) }.

  • Transitive: example in class.

  • Equivalence: {(a,b) in N times N | a = b }.

  • Equivalence-2: {(a,b) in N times N | a mod 7 = b mod 7 }.

What about the empty set as a relation? Is it reflexive? Symmetric? Transitive??


Conider the standard leq relation over mathbb{N}.

 { (i,j) | exists k in mathbb{N} i + k  = j }

  • It is reflexive since i leq i for all i in mathbb{N}.

  • It is not symmetric: 1 leq 2 but 2 notleq 1.

  • It is transitive: a leq b mbox{AND} b leq c Rightarrow a leq c.

  • It is not irreflive since 1 leq 1

Anti-Symmetric Relation

We looked at irreflexive relations as the polar opposite of reflexive (and not just the logical negation). Now we consider a similar concept of anti-symmetric relations.

This is a special property that is not the negation of symmetric.

A relation R is anti-symmetric iff whenever (a,b) and (b,a) are both in R then a=b.

Anti-symmetric is not the opposite of symmetric. A relation can be both symmetric and anti-symmetric:

R = {(1,1), (2,2), (3,3) }

Another example is the empty set. It is both symmetric and anti-symmetric.

The leq relation on N is anti-symmetric. Whenever a leq b and b leq a then a=b. In fact, the notion of anti-symmetry is useful to talk about ordering relations such as subseteq over sets and leq over natural numbers.

Partial Orders

A relation R is a partial order iff it is

  • reflexive

  • transitive

  • anti-symmetric.

The easiest way to remember a partial order is to think of the subseteq relation over sets. In fact, the partial order definition is an abstraction of the subset relation.


Let us take the words in an english dictionary. The lexicographic ordering (or alphabetic ordering) is a partial order.


Is the order leq on N a partial order? What about the < ordering?


Is the = relation on N a partial order?


Is the empty relation a partial order, in general?

Partial orders can leave elements incomparable. We saw the example with subseteq where {1} is incomparable in the subseteq ordering to {2,3}.

A strict partial order is an irreflexive, transitive and anti-symmetric relation. It is inspired by the subset order between sets which captures the notion of proper subsets.

Total Order

A total ordering is a partial order that also satisfies the property that there are no incomparable elements.

The definition of a total ordering is inspired by the leq relation. It is a partial order over N but additionally for any m,n in N either m leq n or n leq m.

A strict total ordering is a strict partial order that ensures that no two different elements are incomparable.

Strict total orderings are inspired by the < relation on N.

More Examples

Let us take an example and study various find of relations. Let us take the set of natural numbers and classify relations over them.

  • R_1 = { (m,n)  | m-n is even }.

  • R_2 = { (m,n)  | n = m^2 } .

  • R_3 = { (m,n)  | m leq n + 2 }.

  • R_4 = { (m,n)  |  n -2 leq m  leq n+2 }.

Relation Reflexive?? Symmetric?? Transitive? Anti-Symmetric? Irreflexive?
1 Y Y Y N N
2 N N N Y N