1. Properties of Relations In many applications to computer science and applied mathematics, we deal with relations on a set A rather than relations from A to B A relation R on a set A is reflexive if (a,a)  R for all a  A; it’s irreflexive if (a,a)  R for all a  A Let  ={(a,a)| a  A},  is the relation of equality on the set A, and  is reflexive The matrix of a reflexive relation must have all 1’s on its main diagonal, while irreflexive must have all 0’s on its main diagonal. A reflexive relation has a cycle of length 1 at every vertex, while irreflexive has no cycles of length 1. R is reflexive if and only if  R, and R is irreflexive if and only if  R=  If R is reflexive on a set A, then Dom(R)=Ran(R)=A.

2. Symmetric, Asymmetric and Antisymmetric Relations A relation R on a set A is symmetric if whenever aRb, then bRa –R is not symmetric if there are some a and b  A with aRb, but bRa A relation R on a set A is asymmetric if whenever aRb, then bRa –R is not asymmetric if there are some a and b  A with both aRb and bRa A relation R on a set A is antisymmetric if whenever aRb and bRa, then a=b; –The contrapositive of this definition: R is antisymmetric if whenever a  b, then aRb or bRa –R is not antisymmetric if there are some a and b  A, a  b, and both aRb and bRa

3. Symmetric, Asymmetric and Antisymmetric Relations M R =[m ij ] of a symmetric relation satisfies the property that if m ij =1, then m ji =1 and if m ij =0, then m ji =0; it’s a symmetric matrix –m ii can be either 0 or 1 for all i M R =[m ij ] of an asymmetric relation satisfies the property that if m ij =1, then m ji =0 and m ii =0 for all i –If m ij =0, then m ji can be either 0 or 1 M R =[m ij ] of an antisymmetric relation satisfies the property that if i  j, then m ij =0, or m ji =0 –m ii can be either 0 or 1 for all i

4. Symmetric, Asymmetric and Antisymmetric Relations If relation R is asymmetric, then the digraph of R cannot simultaneously have an edge from vertex i to j and an edge from j to i; and there can be no cycles of length 1: all edges are one-way If relation R is antisymmetric, then for different vertices i and j there cannot not be an edge from i to j and an edge from j to i; when i=j, no condition is imposed, thus there may be cycles of length 1: still all edges are one-way If relation R is symmetric, then whenever there is an edge from vertex i to j, then there is an edge from vertex j to i.

5. Transitive Relations A relation R on a set A is transitive if whenever aRb and bRc, then aRc. A relation R on A is not transitive if there exists a, b, and c in A so that aRb and bRc, but aRc. –If such a, b, and c do not exist, then R is transitive R is transitive if and only if its matrix M R =[m ij ] has the property if m ij =1 and m jk =1, then m ik =1. If M R  M R has a 1 in any position, then M R must have a 1 in the same position, i.e. if M R  M R = M R, then R is transitive –The converse is not true, i.e. if M R  M R  M R, then R may be transitive or may not be transitive

6. Transitive Relations Theorem 1. A relation R is transitive if and only if it satisfies the following property: if there is a path of length greater than 1 from vertex a to vertex b, there is a path of length 1 from a to b (that is, a is related to b). Algebraically stated, R is transitive if and only if R n  R for all n  1. Theorem 2. Let R be a relation on a set A, then (a)Reflexive of R means that a  R(a) for all a in A. (b)Symmetry of R means that a  R(b) if and only if b  R(a). (c)Transitivity of R means that if b  R(a) and c  R(b), then c  R(a).

7. A relation R on a set A is called an equivalence relation if it is reflexive, symmetric and transitive. Theorem 1. Let P be a partition of a set A. Recall that the sets in P are called the blocks of P. Define the relation R on A as follows: aRb if only if a and b are members of the same block. Then R is an equivalence relation on A. R will be called the equivalence relation determined by P –each element in a block is related to every element in the same block and only to these elements. Lemma 1. Let R be an equivalence relation on A, and let a  A and b  A, then aRb if and only if R(a)=R(b). Equivalence Relations and Partitions

8. Theorem 2. Let R be an equivalence relation on A, and let P be the collection of all distinct relative sets R(a) for a in A. Then P is a partition of A and R is the equivalence relation determined by P. If R is an equivalence relation on A, then the sets R(a) are called equivalence classes of R. The partition P constructed in Theorem 2 consists of all equivalence classes of R, denoted by A/R P is the quotient set of A, that is constructed from and determines R.

9. Equivalence Relations and Partitions A general procedure for determining partitions A/R for a set A which is finite or countable: step1: choose any element a of A and compute the equivalence class R(a) step2: if R(a)  A, choose an element b, not included in R(a), and compute the equivalence class R(b) step3: if A is not the union of previously computed equivalence classes, then choose an element x of A that is not in any of those equivalence classes and compute R(x) step4: repeat step3 until all element of A are included in the computed equivalence classes. If A is countable, this process could continue indefinitely. In that case, continue until a pattern emerges that allows to describe or give a formula for all equivalence classes.