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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 aA; it’s irreflexive if (a,a)R for all aA • Let ={(a,a)| aA}, 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. 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 bA 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 bA 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 ab, then aRb or bRa – R is not antisymmetric if there are some a and bA, ab, and both aRb and bRa Symmetric, Asymmetric and Antisymmetric Relations • MR =[mij] of a symmetric relation satisfies the property that if mij=1, then mji=1 and if mij=0, then mji=0; it’s a symmetric matrix – mii can be either 0 or 1 for all i • MR =[mij] of an asymmetric relation satisfies the property that if mij=1, then mji=0 and mii=0 for all i – If mij=0, then mji can be either 0 or 1 • MR =[mij] of an antisymmetric relation satisfies the property that if ij, then mij=0, or mji=0 – mii can be either 0 or 1 for all i 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. 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 MR =[mij] has the property if mij=1 and mjk=1, then mik=1. If MRMR has a 1 in any position, then MR must have a 1 in the same position, i.e. if MRMR = MR, then R is transitive – The converse is not true, i.e. if MRMR MR, then R may be transitive or may not be transitive 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 RnR for all n1. Theorem 2. Let R be a relation on a set A, then (a) Reflexive of R means that aR(a) for all a in A. (b) Symmetry of R means that aR(b) if and only if bR(a). (c) Transitivity of R means that if bR(a) and cR(b), then cR(a) . Equivalence Relations and Partitions • 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 aA and bA, then aRb if and only if R(a)=R(b). Equivalence Relations and Partitions 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. 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.