Properties of Relations In many applications to computer science and applied mathematics, we deal...
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Transcript of Properties of Relations In many applications to computer science and applied mathematics, we deal...
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) .
• 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
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.