Relations Handout
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Transcript of Relations Handout
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RELATIONS
RELATIONS
RICKY F. RULETE
Department of Mathematics and StatisticsUniversity of Southeastern Philippines
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RELATIONS
PRODUCT SETS
Product Sets
An ordered pair of elements a and b, where a is designated asthe first element and b as the second element, is denoted by(a, b). In particular,
(a, b) = (c, d)
if and only if a = c and b = d. Thus (a, b) 6= (b, a). Thiscontrasts with sets where the order of elements is irrelevant; forexample, {3, 5} = {5, 3}.
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RELATIONS
PRODUCT SETS
Product Sets
Consider two arbitrary sets A and B. The set of all orderedpairs (a, b) where a A and b B is called the product, orCartesian product, of A and B and is denoted by AB. Bydefinition,
AB = {(a, b) | a A and b B}
One frequently writes A2 instead of AA.
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RELATIONS
PRODUCT SETS
Product Sets
Example (2.1)
R2 = R R is the set of ordered pairs of real numbers.
Example (2.2)
Let A = {1, 2} and B = {a, b, c}. Then
AB = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}B A = {(a, 1), (b, 1), (c, 1), (a, 2), (b, 2), (c, 2)}
Also, AA = {(1, 1), (1, 2), (2, 1), (2, 2)}.
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RELATIONS
PRODUCT SETS
Product Sets
Remark
1 AB 6= B A2 For any finite sets A and B, n(AB) = n(A)n(B). This
follows from the observation that, for an ordered pair (a, b)in AB, there are n(A) possibilities for a, and for each ofthese there are n(B) possibilities for b.
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RELATIONS
PRODUCT SETS
Product Sets
For any sets A1, A2, . . . , An, the set of all ordered n-tuples(a1, a2, . . . , an) where a1 A1, a2 A2, . . . , an An is calledthe product of the sets A1, A2, . . . , An and is denoted by
A1 A2 An orni=1
Ai
We write An instead of AA A, where there are nfactors all equal to A. For example, R3 = R R R.
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RELATIONS
RELATIONS
Relations
Definition (2.1)
Let A and B be sets. A binary relation or, simply, relationfrom A to B is a subset of AB.
Suppose R is a relation from A to B. Then, for each pair a Aand b B, exactly one of the following is true:
1 (a, b) R; we then say a is R-related to b, written aRb.2 (a, b) / R; we then say a is not R-related to b, writtenaupslopeRb.
If R is a relation from a set A to itself, that is, R is a subsetof A2 = AA, then we say that R is a relation on A.
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RELATIONS
RELATIONS
Relations
The domain of a relation R is the set of all first elements of theordered pairs which belong to R, and the range is the set ofsecond elements.
Example (2.3)
1 Let A = {1, 2, 3} and B = {x, y, z} and letR = {(1, y), (1, z), (3, y)}. Then R is a relation from A to Bsince R is a subset of AB. With respect to this relation,
1Ry, 1Rz, 3Ry, but 1upslopeRx, 2upslopeRx, 2upslopeRy, 2upslopeRz, 3upslopeRx, 3upslopeRz
The domain of R is {1, 3} and the range is {y, z}.
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RELATIONS
RELATIONS
Relations
Example (2.3)
2 Set inclusion is a relation on any collection of sets. For,given any pair of sets A and B, either A B or A 6 B.
3 A familiar relation on the set Z of integers is m dividesn. A common notation for this relation is to write m|nwhen m divides n. Thus 6|30 but 7 - 25.
4 Consider the set L of lines in the plane. Perpendicularity,written , is a relation on L. That is,given any pair oflines a and b, either a b or a 6 b. Similarly, is parallelto, written , is a relation on L since either a b ora b.
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RELATIONS
RELATIONS
Relations
Example (2.3)
5 Let A be any set. An important relation on A is that ofequality,
{(a, a) | a A}which is usually denoted by =. This relation is also
called the identity or diagonal relation on A and it will bealso denoted by A or simply .
6 Let A be any set. Then AA and are subsets of AAand hence are relations on A called the universal relationand empty relation, respectively.
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RELATIONS
RELATIONS
Inverse Relations
Let R be any relation from a set A to a set B. The inverse ofR, denoted by R1, is the relation from B to A which consistsof those ordered pairs which, when reversed, belong to R; thatis,
R1 = {(b, a) | (a, b) R}For example, let A = {1, 2, 3} and B = {x, y, z}. Then the
inverse ofR = {(1, y), (1, z), (3, y)}
isR1 = {(y, 1), (z, 1), (y, 3)}
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RELATIONS
RELATIONS
Inverse Relations
Remark
1 If R is any relation, then (R1)1 = R.2 The domain and range of R1 are equal, respectively, to
the range and domain of R.
3 If R is a relation on A, then R1 is also a relation on A.
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RELATIONS
PICTORIAL REPRESENTATIVES OF RELATIONS
Relations on R
Let S be a relation on R of real numbers; that is, S is a subsetof R2 = R R. Frequently, S consists of all ordered pairs ofreal numbers which satisfy some given equation E(x, y) = 0(such as x2 + y2 = 25). The pictorial representation of therelation is sometimes called the graph of the relation. Forexample, the graph of the relation x2 + y2 = 25 is a circlehaving its center at the origin and radius 5.
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RELATIONS
PICTORIAL REPRESENTATIVES OF RELATIONS
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x2 + y2 = 25
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RELATIONS
PICTORIAL REPRESENTATIVES OF RELATIONS
Consider the relation R on the set A = {1, 2, 3, 4}:
R = {(1, 2), (2, 2), (2, 4), (3, 2), (3, 4), (4, 1), (4, 3)}
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RELATIONS
PICTORIAL REPRESENTATIVES OF RELATIONS
Pictures of Relations on Finite Sets
Suppose A and B are finite sets. There are two ways ofpicturing a relation R from A to B.
1 Form a rectangular array (matrix) whose rows are labeledby the elements of A and whose columns are labeled by theelements of B. Put 1 or 0 in each position of the arrayaccording as a A is or is not related to b B. This arrayis called the matrix of the relation.
2 Write down the elements of A and the elements of B in twodisjoint disks, and then draw an arrow from a A to b Bwhenever a is related to b. This picture will be called thearrow diagram of the relation.
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RELATIONS
PICTORIAL REPRESENTATIVES OF RELATIONS
Pictures of Relations on Finite Sets
R = {(1, y), (1, z), (3, y)}
x y z
1 0 1 12 0 0 03 0 1 0
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RELATIONS
PICTORIAL REPRESENTATIVES OF RELATIONS
Pictures of Relations on Finite Sets
R = {(1, y), (1, z), (3, y)}
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1
2
3
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y
z
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RELATIONS
COMPOSITION OF RELATIONS
Composition of Relations
Let A, B and C be sets, and let R be a relation from A to Band let S be a relation from B to C. That is, R is a subset ofAB and S is a subset of B C. Then R and S give rise to arelation from A to C denoted by R S and defined by:
a(R S)c if for some b B we have aRb and bSc.
That is,
R S = {(a, c) | there exists b B for which (a, b) Rand (b, c) S}
The relation R S is called composition of R and S.
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RELATIONS
COMPOSITION OF RELATIONS
Composition of Relations
Remark
1 If R is a relation on a set A, then R R , the compositionof R with itself, is always defined.
2 R R is sometimes denoted by R2. Similarly,R3 = R2 R = R R R. Thus Rn is defined for allpositive n.
3 Many texts denote the composition of relations R and Sby S R rather than R S.
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RELATIONS
COMPOSITION OF RELATIONS
Composition of Relations
Example (2.4)
Let A = {1, 2, 3, 4}, B = {a, b, c, d}, C = {x, y, z} and let
R = {(1, a), (2, d), (3, a), (3, b), (3, d)}
andS = {(b, x), (b, z), (c, y), (d, z)}
Then R S = {(2, z), (3, x), (3, z)}.
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RELATIONS
COMPOSITION OF RELATIONS
Composition of Relations
Theorem (2.1)
Let A, B, C and D be sets. Suppose R is a relation from A toB, S is a relation from B to C, and T is a relation from C toD. Then
(R S) T = R (S T )
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RELATIONS
COMPOSITION OF RELATIONS
Composition of Relations
Proof.
Suppose (a, d) belongs to (R S) T . Then there exists c Csuch that (a, c) R S and (c, d) T . Since (a, c) R S,there exists b B such that (a, b) R and (b, c) S. Since(b, c) S and (c, d) T , we have (b, d) S T ; and since(a, b) R and (b, d) S T , we have (a, d) R (S T ).Therefore, (R S) T R (S T ). SimilarlyR (S T ) (R S) T . Both inclusion relations prove(R S) T = R (S T ).
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RELATIONS
COMPOSITION OF RELATIONS
Composition of Relations and Matrices
Let MR and MS denote respectively the matrix representationof the relations R and S. Then
MR =
1 0 0 00 0 0 11 1 0 10 0 0 0
and MS =
0 0 01 0 10 1 00 0 1
Multiplying MR and MS we obtain the matrix
MRMS =
0 0 00 0 11 0 20 0 0
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RELATIONS
TYPES OF RELATIONS
Reflexive Relations
A relation R on a set A is reflexive if aRa for every a A,that is, if (a, a) R for every a A. Thus R is not reflexive ifthere exists a A such that (a, a) / R.
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RELATIONS
TYPES OF RELATIONS
Reflexive Relations
Example (2.5)
Consider the following five relations on the set A = {1, 2, 3, 4}:
R1 = {(1, 1), (1, 2), (2, 3), (1, 3), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}R3 = {(1, 3), (2, 1)}R4 = , the empty relationR5 = AA, the universal relation
Determine which of the relations are reflexive.
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RELATIONS
TYPES OF RELATIONS
Reflexive Relations
Example (2.6)
Consider the following five relations:
1 Relation (less than or equal) on the set Z of integers.2 Set inclusion on a collection C of sets.3 Relation (perpendicular) on the set L of lines in the
plane.
4 Relation (parallel) on the set L of lines in the plane.5 Relation | of divisibility on the set Z+ of positive integers.
(Recall x | y if there exists z such that xz = y.)Determine which of the relations are reflexive.
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RELATIONS
TYPES OF RELATIONS
Symmetric Relations
A relation R on a set A is symmetric if whenever aRb thenbRa, that is, if whenever (a, b) R then (b, a) R. Thus R isnot symmetric if there exists a, b A such that (a, b) R but(b, a) / R.
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RELATIONS
TYPES OF RELATIONS
Symmetric Relations
Example (2.7)
1 Determine which of the following five relations on the setA = {1, 2, 3, 4} are symmetric.
R1 = {(1, 1), (1, 2), (2, 3), (1, 3), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}R3 = {(1, 3), (2, 1)}R4 = , the empty relationR5 = AA, the universal relation
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RELATIONS
TYPES OF RELATIONS
Symmetric Relations
Example (2.7)
2 Determine which of the following five relations aresymmetric.
1 Relation (less than or equal) on the set Z of integers.2 Set inclusion on a collection C of sets.3 Relation (perpendicular) on the set L of lines in the
plane.4 Relation (parallel) on the set L of lines in the plane.5 Relation | of divisibility on the set Z+ of positive integers.
(Recall x | y if there exists z such that xz = y.)
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RELATIONS
TYPES OF RELATIONS
Antisymmetric Relations
A relation R on a set A is antisymmetric if whenever aRb andbRa then a = b, that is, if a 6= b and aRb then bupslopeRa. Thus R isnot antisymmetric if there exists distinct elements a and b in Asuch that aRb and bRa.
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RELATIONS
TYPES OF RELATIONS
Antisymmetric Relations
Example (2.8)
1 Determine which of the following five relations on the setA = {1, 2, 3, 4} are antisymmetric.
R1 = {(1, 1), (1, 2), (2, 3), (1, 3), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}R3 = {(1, 3), (2, 1)}R4 = , the empty relationR5 = AA, the universal relation
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RELATIONS
TYPES OF RELATIONS
Antisymmetric Relations
Example (2.8)
2 Determine which of the following five relations areantisymmetric.
1 Relation (less than or equal) on the set Z of integers.2 Set inclusion on a collection C of sets.3 Relation (perpendicular) on the set L of lines in the
plane.4 Relation (parallel) on the set L of lines in the plane.5 Relation | of divisibility on the set Z+ of positive integers.
(Recall x | y if there exists z such that xz = y.)
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RELATIONS
TYPES OF RELATIONS
Remark
The properties of being symmetric and being antisymmetricare not negatives of each other. For example, the relationR = {(1, 3), (3, 1), (2, 3)} is neither symmetric norantisymmetric. On the other hand, the relationR = {(1, 1), (2, 2)} is both symmetric and antisymmetric.
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RELATIONS
TYPES OF RELATIONS
Transitive Relations
A relation R on a set A is transitive if whenever aRb and bRcthen aRc , that is, if whenever (a, b), (b, c) R then (a, c) R.Thus R is not transitive if there exists a, b, c A such that(a, b), (b, c) R but (a, c) / R.
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RELATIONS
TYPES OF RELATIONS
Transitive Relations
Example (2.9)
1 Determine which of the following five relations on the setA = {1, 2, 3, 4} are transitive.
R1 = {(1, 1), (1, 2), (2, 3), (1, 3), (4, 4)}R2 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}R3 = {(1, 3), (2, 1)}R4 = , the empty relationR5 = AA, the universal relation
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RELATIONS
TYPES OF RELATIONS
Transitive Relations
Example (2.9)
2 Determine which of the following five relations aretransitive.
1 Relation (less than or equal) on the set Z of integers.2 Set inclusion on a collection C of sets.3 Relation (perpendicular) on the set L of lines in the
plane.4 Relation (parallel) on the set L of lines in the plane.5 Relation | of divisibility on the set Z+ of positive integers.
(Recall x | y if there exists z such that xz = y.)
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations
Consider a nonempty set S. A relation R on S is anequivalence relation if R is reflexive, symmetric, and transitive.That is, R is an equivalence relation on S if it has the followingthree properties:
1 For every a S, aRa.2 If aRb, then bRa.
3 If aRb and bRc, then aRc.
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations
The general idea behind an equivalence relation is that it is aclassification of objects which are in some way alike. In fact,the relation = of equality on any set S is an equivalencerelation; that is
1 a = a for every a S.2 If a = b, then b = a.
3 If a = b and b = c, then a = c.
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations
Example (2.10)
1 Let L be the set of lines and let T be the set of triangles inthe Euclidean plane.
The relation is parallel to or identical to is an equivalencerelation on L.The relations of congruence and similarity are equivalencerelations on T .
2 The relation of set inclusion is not an equivalencerelation. It is not symmetric.
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations and Partitions
Suppose R is an equivalence relation on a set S. For eacha S, let [a] denote the set of elements of S to which a isrelated under R; that is:
[a] = {x | (a, x) R}
We call [a] the equivalence class of a in S; any b [a] is called arepresentative of the equivalence class. The collection of allequivalence classes of elements of S under an equivalencerelation R is denoted by S/R, that is,
S/R = {[a] | a S}
It is called the quotient set of S/R.
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations and Partitions
Theorem (2.2)
Let R be an equivalence relation on a set S. Then S/R is apartition of S. Specifically:
1 For each a in S, we have a [a].2 [a] = [b] if and only if (a, b) R.3 If [a] 6= [b], then [a] and [b] are disjoint.
Conversely, given a partition {Ai} of the set S, there is anequivalence relation R on S such that the sets Ai are theequivalence classes.
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations and Partitions
Proof
1 Since R is reflexive, (a, a) R for every a A andtherefore a [a].
2 Suppose (a, b) R. We want to show that [a] = [b]. Letx [b]; then (b, x) R. But by hypothesis (a, b) Rand so, by transitivity, (a, x) R. Accordingly x [a].Thus [b] [a]. To prove that [a] [b] we observe that(a, b) R implies, by symmetry, that (b, a) R. Then, bya similar argument, we obtain [a] [b]. Consequently,[a] = [b]. On the other hand, if [a] = [b], then by (1),b [b] = [a]; hence (a, b) R.
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations and Partitions
Proof
3 We prove that equivalent contrapositive statement:
If [a] [b] 6= then [a] = [b]
If [a] [b] 6= then there exists an element x A withx [a] [b]. Hence (a, x) R and (b, x) R. Bysymmetry, (x, b) R and by transitivity, (a, b) R.Consequently by (2), [a] = [b].
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RELATIONS
EQUIVALENCE RELATIONS
Equivalence Relations and Partitions
Example (2.11)
Consider the relation R = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3)} onS = {1, 2, 3}. Observe that R is an equivalence relation. Also:
[1] = {1, 2}, [2] = {1, 2}, [3] = {3}
Observe that [1] = [2] and that S/R = {[1], [3]} is a partitionof S.
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RELATIONS
EQUIVALENCE RELATIONS
Partial Ordering Relations
A relation R on a set S is called partial ordering or a partialorder of S if R is reflexive, antisymmetric, and transitive. A setS together with a partial ordering R is called a partially orderedset or poset.
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RELATIONS
EQUIVALENCE RELATIONS
Partial Ordering Relations
Example (2.12)
1 The relation of set inclusion is a partial ordering on anycollection of sets since set inclusion has the three desiredproperties. That is,
A A for any set A.If A B and B A, then A = B.If A B and B C, then A C.
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RELATIONS
EQUIVALENCE RELATIONS
Partial Ordering Relations
Example (2.12)
2 The relation on the set R of real numbers is reflexive,antisymmetric, and transitive. Thus is a partialordering on R.
3 The relation a divides b, written a | b, is a partialordering on the set Z+ of positive integers. However, adivides b is not a partial ordering on the set Z of integerssince a | b and b | a need not imply a = b. For example,3 | 3 and 3 | 3 but 3 6= 3.
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RELATIONS
EQUIVALENCE RELATIONS
THANK YOU!!!
PRODUCT SETSRELATIONSPICTORIAL REPRESENTATIVES OF RELATIONSCOMPOSITION OF RELATIONSTYPES OF RELATIONSEQUIVALENCE RELATIONS