Relations: Operations

1
Relations Operations Properties
2
Power Sets

• Set of all subsets of a set A.
• A 1,2
• P(A) 2A , 1, 2, 1,2
• We note that each element of A is either present
  (1) or not present (0). If we treat the elements
  of A as a sequence, we get a bit-string that
  represents the set.
• If A a,b,c,d, we can order the elements in a
  sequence lta,b,c,dgt.
• Then, we can let the bit-string say which
  elements are present e.g. 0110 means b,c.
• We can represent all the subsets of A, from ?
  0000 to U 1111.
• This bit-string notation also helps us know the
  number of subsets in the powerset (just count in
  binary)
• 2A, 2A 2 2 2 (A times)
• This motivates the notation 2A for the power set.

3
Bit-String Operations on Power Sets

• With bit string representations
• Set intersection ? pairwise ?
• Set union ? pairwise ?
• Set complement bit complement
• Set minus mask out using 1s complement
  2nd operand and do pairwise ?
• E.g. using a,b,c,d
• 1011 ? 1101 1001 i.e. a,c,d ? a,b,d a,d
• 1011 ? 1101 1111 i.e. a,c,d ? a,b,d
  a,b,c,d
• 1011 0100 i.e. a,c,d b
• 1010 1100 0010 i.e. a,c a,b c

4
Binary Relations

• Sets of ordered 2-tuples (pairs) with values
  selected from domains (sets)
• Formally, a relation R from set A to set B is a
  set of pairs (x,y) such that x ? A and y ? B. We
  may write RA ? B to express this.
• R ? A?B
• If (x,y) ? R, we say that x is R-related to y we
  may also write xRy.
• Example
• lt N?N (where N is the set of natural numbers)
• (2,3) ? lt or 2lt3
• Predicates also define relations
• lt0,1,2??1,0,1,2 (x,y) x ? 0,1,2 ? y ?
  ?1,0,1,2 ? x lt y
• (0,1), (0,2), (1,2)

5
Representations for Binary Relations
lt -1 0 1 2
0 0 0 1 1
1 0 0 0 1
2 0 0 0 0

• Tables

-1 0 1 2
0 0 0 1 1
1 0 0 0 1
2 0 0 0 0
Matrices
Graphs - directed graph or digraph - directed arcs
6
Domains and Ranges for Binary Relations

• Let R (0,1), (0,2), (1,2), then
• Domain of R dom R dom(R) x ?y((x,y) ?
  R)
• i.e. 0,1
• like ?XR
• Range of R ran R ran(R) y ?x((x,y) ? R)
• i.e. 1,2
• like ?YR

7
Inverse

• If R A?B, then the inverse of R is R B?A.
• R-1 is also a common notation
• R is defined by (y,x) (x,y) ? R.
• If R (a,b), (a,c), then R (b,a), (c,a)
• gt is the inverse of lt on the reals
• Note that R ? R
• The complement of lt is ?
• But the inverse of lt is gt

8
Set Operations

• Since relations are sets, set operations apply
  (just like relational algebra).
• The arity must be the same ? indeed, the sets for
  the domain space and range space must be the same
  (just like in relational algebra).

9
Restriction Extension

• Restriction decreases the domain space or range
  space.
• Example Let lt be the relation (1,2), (1,3),
  (2,3). The restriction of the domain space to
  1 restricts lt to (1,2), (1,3).
• Extensions increase the domain space or range
  space.
• Example Let lt be the relation (1,2), (1,3),
  (2,3). The extension of both the domain space
  and range space to 1,2,3,4 extends lt to (1,2),
  (1,3), (1,4), (2,3), (2,4), (3,4)
• Extensions and restrictions need not change the
  relation.
• Suppose M1, M2, M3 are three males and F1, F2,
  F3 are three females and married_to is (M1,F3),
  (M2,F2).
• Removing the unmarried male M3 or the unmarried
  female F1 does nothing to the relation, and
  adding the unmarried male M4 or the unmarried
  female F4 does nothing to the relation.

10
Composition

• Let RA?B and SB?C be two relations. The
  composition of R and S, denoted by R?S, contains
  the pairs (x,z) if and only if there is an
  intermediate object y such that (x,y) is in R and
  (y,z) is in S.
• Given RA?B and SB?C, with A 1,2,3,4, B
  1,2,3,4,5, C 1,2,3,4, and R (1,3),
  (4,2), (1,1) and S (3,4), (2,1), (4,2), we
  have
• R?S (1,4), (4,1)
• S?R (3,2), (2,3), (2,1) (i.e. not
  commutative)
• R?(S?R) (1,2), (4,3), (4,1)
• (R?S)?R (1,2), (4,3), (4,1) (i.e. is
  associative)

1234
12345
1234
11
Composition Matrix Multiplication

• Relation matrices only contain 0s and 1s.
• For relational matrix multiplication, use ? for
  , ? for , with 1T and 0F.

?

R?S(1,1) (1?0)?(0?1)?(1?0)?(0?0)?(0?0) 0
R?S (1,4), (4,1)
12
Combined Operations

• All operations we have discussed can be combined
  so long as the compatibility requirements are
  met.
• Example
• R (1,1), (3,1), (1,2)
• S (1,1), (2,3)
• (R ? S) ? R
• (1,1), (3,1), (1,2), (2,3) ? R
• (1,1), (1,3), (2,1), (3,2) ? R
• (1,1), (1,2), (2,1), (2,2)

R A ? B S B ? A A 1,2,3 B 1,2,3
13
Properties of Binary Relations

• Properties of binary relations on a set R A?A
  help us with lots of things ? groupings,
  orderings,
• Because there is only one set A
• matrices are square A ? A
• graphs can be drawn with only one set of nodes

R (1,3), (4,2), (3,3), (3,4)
14
Reflexivity

• Reflexive ?x(xRx)
• , ?, is in same major as, etc.

Reflexive
Note
Irreflexive
Neither
e.g., loves
e.g., lt
15
Symmetry

• Symmetric ?x?y(xRy ? yRx)
• Antisymmetric ?x?y(xRy ? yRx ? x y)
• sibling is symmetric
• ?, ? are antisymmetric (if a?b and b?a, then
  ab)
• is symmetric and antisymmetric
• loves is neither symmetric nor antisymmetric

Note antisymmetry is NOT asymmetry
16
Transitivity

• Transitive ?x?y?z(xRy ? yRz ? xRz)
• taller is transitive
• lt is transitive xltyltz
• ancestor is transitive
• brother-in-law is not transitive
• Transitive If I can get there in two, then I
  can get there in one.

17
Transitivity (contd)

• Note xRy ? yRz corresponds to xR?Rz, which is
  equivalent to xRRz xR2z thus we can define
  transitivity as
• ?x?y?z(xR2z ? xRz)
• If I can get there in two, then I can get there
  in one.
• For transitivity
• if reachable through an intermediate, then
  reachable directly is required.
• if (x,z)?R2, then (x,z)?R
• ? (x,z)?R2 ? (x,z)?R
• ? R2 ? R (recall our def. of subset x?A ? x?B,
  then A?B)

18
Transitivity (contd)
?

R
R2 ? R
R
Transitive
?

R
R
R2 ? R
Not Transitive
19
Closures Equivalence Relations
20
Closure

• Closure means adding something until done.
• Normally adding as little as possible until some
  condition is satisfied
• Least fixed point similarities

21
Reflexive Closure

• Reflexive closure of a relation R(r)
• smallest reflexive relation that contains R (i.e.
  fewest pairs added)
• R(r) R ? IA (R is a relation on a set A,
  and IA is the identity relation ? 1s on
  the diagonal and 0s elsewhere.)

R
R(r)
?x (xRx)
22
Symmetric Closure

• Symmetric closure of a relation R(s)
• smallest symmetric relation that contains R (i.e.
  fewest pairs added)
• R(s) R ? R (R is R inverse)

R
R
R ? R
?x?y(xRy ? yRx)
23
Transitive Closure

• Transitive closure of a relation R(t) R
• smallest transitive relation that contains R
  (i.e. fewest pairs added)
• for each path of length i, there must be a direct
  path. (This follows from x?y, y?z ? x?z since,
  if we also have v?x, we must have v?z, a path of
  length 3.)
• R(t) R ? R2 ? R3 ? ? RA. (No path can be
  longer than A, the number of elements in A.)

24
Transitive Closure Example 1
1
All paths of length 1
R
2
3
1
All paths of length 2
R2
2
3
1
All paths of length 3
RR2 R3
2
3
R?R2?R3
25
Transitive Closure Example 2
1
R
All paths of length 1
2
3
1
R2
All paths of length 2
2
3
1
RR2 R3
All paths of length 3
2
3
1
R?R2?R3
Paths of any length
2
3
26
Reflexive Transitive Closure

• Reflexive transitive closure of a relation R
• smallest reflexive and transitive relation that
  contains R
• R IA ? R R0 ? R R0 ? R1 ? R2 ? RA
• Example

IA R0
R1 ? R2 ? R3
1
R0 ? R1 ? R2 ? R3
2
3
27
Equivalence Relations

• A relation R on a set A is an equivalence
  relation if R is reflexive, symmetric, and
  transitive.
• Equivalence relations are about equivalence
• Examples
• for integers x x reflexive
• x y ? y x symmetric
• xy ? yz ? xz transitive
• for sets A A reflexive
• A B ? B A symmetric
• AB ? BC ? AC transitive
• Let R be has same major as for college students
• xRx ? reflexive same major as self
• xRy ? yRx ? symmetric same major as each
  other
• xRy ? yRz ? xRz ? transitive same as, same as
  ? same as

28
Partitions

• A partition of a set S is
• a set of subsets Si1,2,n of S
• such that ?ni1 Si S, Sj ? Sk ? for j ? k.
• Each Si is called a block (also called an
  equivalence class)
• Example
• Suppose we form teams (e.g. for a doubles tennis
  tournament) from the set
• Abe, Kay, Jim, Nan, Pat, Zed
• then teams could be
• Abe, Nan, Kay, Jim, Pat, Zed
• Note on same team as is reflexive, symmetric,
  transitive ? an equivalence relation.
  Equivalence relations and partitions are the same
  thing (two sides of the same coin).

29
Partitions (contd)

• Since individual elements can only appear in one
  block (Sj ? Sk ? for j ? k), blocks can be
  represented by any element within the block.
• e.g. Nans Team
• Jimmers 2011 sweet-16 team
• Formally, x set of all elements related to x
  and y ? x iff xRy
• e.g. Nan represents Abe, Nan, Nans team
• Abe represents Abe, Nan, Abes team
• Jimmer represents the set of players who
  played on BYUs 2011 sweet-16 team

30
Partitions Equivalence Relations
Example

• The mod function partitions the natural numbers
  into equivalence classes.
• 0 mod 3 0 so 0 forms a class 0
• 1 mod 3 1 so 1 forms new class 1
• 2 mod 3 2 so 2 forms new class 2
• 3 mod 3 0 so 3 belongs to 0
• 4 mod 3 1 so 4 belongs to 1
• 5 mod 3 2 so 5 belongs to 2
• 6 mod 3 0 so 6 belongs to 0
• Thus, the mod function partitions the natural
  numbers into equivalence classes.
• 0 0, 3, 6,
• 1 1, 4, 7,
• 2 2, 5, 8,

31
Partitions ? Equivalence Relations
Theorem If S1, , Sn is a partition of S, then
RS?S is an equivalence relation, where R is in
same block as. Note to prove that R is an
equivalence relation, we must prove that R is
reflexive, symmetric, and transitive. Proof
Reflexive since every element is in the same
block as itself. Symmetric since if x is in the
same block as y, y is in the same block as x.
Transitive since if x and y are in the same
block and y and z are in the same block, x and z
are in the same block.
32
Equivalence Relations ? Partitions
Theorem If RS?S is an equivalence relation and
x y xRy , then x x ? S is a
partition P of S. Note to prove that we have a
partition, we must prove (1) that every element
of S is in a block of P, and (2) that for every
pair of distinct blocks Sj and Sk (j?k) of P, Sj
? Sk ?. Proof (1) Since R is reflexive, xRx,
every element of S is at least in its own block
and thus in some block of P. (2) Suppose Sj ? Sk
? ? for distinct blocks Sj and Sk of P. Then, at
least one element y is in both Sj and Sk. Let Sj
y, x1, xn and Sk y, z1, zm, then
yRxi, i1, 2, , n, and yRzp, p1, 2, , m.
Since R is symmetric, xiRy, and since R is
transitive xiRzp. But now, since the elements of
Sj are R-related to the elements of Sk, x1, ,
xn, y, z1, , zm are together in a block of P and
thus Si and Sk are not distinct blocks of P.
33
Partial Orders
34
Partial Orders

• Total orderings single sequence of elements
• Partial orderings some elements may come
  before/after others, but some need not be ordered
• Examples of partial orderings

must be completed before
set inclusion, ?
foundation
framing
plumbing wiring
finishing
a, b, c
a, b a, c b, c
a b c
?
35
Partial Order Definitions(Poset Definitions)

• A relation R S?S is called a (weak) partial
  order if it is reflexive, antisymmetric, and
  transitive.

e.g. ? on the integers

e.g. lt on the integers

• A relation R S?S is called a strict partial
  order if it is irreflexive, antisymmetric, and
  transitive.

36
Total Order

• A total ordering is a partial ordering in which
  every element is related to every other element.
  (This forces a linear order or chain.)
• Examples

1 2 3 4 5
R ? on 1, 2, 3, 4, 5 is total. Pick any two
theyre related one way or the other with respect
to ?.
R ? on a, b, a, b, ? is not total. We
can find a pair not related one way or the other
with respect to ?. a b neither a ? b
nor b ? a
a,b
a
b
?
37
Hasse Diagrams

• We produce Hasse Diagrams from directed graphs
  of relations by doing a transitive reduction plus
  a reflexive reduction (if weak) and (usually)
  dropping arrowheads (using, instead, above to
  give direction)
• 1) Transitive reduction ? discard all arcs except
  those that directly cover an element.
• 2) Reflexive reduction ? discard all self loops.

For ?
we write
a, b
?
b
a
?
38
Upper and Lower Bounds

• If a poset is built from relation R on set A,
  then any x ? A satisfying xRy is an upper bound
  of y, and any x ? A satisfying yRx is a lower
  bound of y.
• Examples If A a, b, c and R is ?, then a,
  c
• - is an upper bound of a, c, and ?.
• - is also an upper bound of a, c (weak poset).
• - is a lower bound of a, b, c.
• - is also a lower bound of a, c (weak poset).

39
Maximal and Minimal Elements

• If a poset is built from relation R on set A,
  then y ? A is a maximal element if there is no x
  such that xRy, and x ? A is a minimal element if
  there is no y such that xRy. (Note We either
  need the poset to be strict or x ? y.)
• In a Hasse diagram, every element with no element
  above it is a maximal element, whereas every
  element with no element below it is a minimal
  element.

Maximal elements
Minimal elements
40
Least Upper and Greatest Lower Bounds

• A least upper bound of two elements x and y is a
  minimal element in the intersection of the upper
  bounds of x and y.
• A greatest lower bound is a maximal element in
  the intersection of the lower bounds of x and y.
• Examples
• For ?, a, c is a least upper bound of a and
  c,
• ? is a greatest lower bound of a and b, c,
  and
• a is a least upper bound of a and ?.
• For the following strict poset, lub(x,y) a,b,
  lub(y,y) a,b,c, lub(a,y) ?, glb(a,b)
  x,y, glb(a,c) y

a
b
c
x
y
41
Functions
42
Function Definition

• A function is a special kind of binary relation.
• A binary relation f ? A ? B is a function if for
  each a ? A there is a unique b ? B

y
x
43
NOT Functions
f (1, a), (2, ß) For each violated
y
Some xs do not have corresponding ys
x
44
NOT Functions
f (1, a), (2, ß), (3, ß), (3, ?) uniqueness
violated for 3 ? appears twice
y
Uniqueness violated for some xs
x
45
Functionswith N-Dimensional Domains

• An (n1)-ary relation f ? A1 ? A2 ? ? An ? B
  is a function if for each lt a1, a2, , angt ? A1 ?
  A2 ? ? An there is a unique b ? B.

46
Notation for Functions

• We can use various notation for functions for f
  (1, a),(2, ß),(3, ß)

Notation (x, y) ? f f x?y y f(x)
Example (2, ß) ? f f 2?ß ß f(2)

• In the notation, x is the argument or preimage
  and y is the image. We can also have the image
  of a set of arguments.
• For functions with n-ary domains, use ltx0, x1, ,
  xngt in place of x.

47
Function Domain and Range

• f A ? B
• A is the domain space
• same as the domain (since all elements
  participate)
• dom f, dom(f), or domain(f)
• B is the range space
• may or may not be the same as the range, which
  is
• y ?x(yf(x))
• All rhs values in pairs (all that get hit)
• ?Bf
• ran f, ran(f), range(f)
• f D1 ? D2 ? ? Dn ? Z
• f Dn ? Z (when all domains are the same)

48
Partial Functions

• Remove the requirement that each a ? A must
  participate. Retain the uniqueness requirement.

Partial Function
Partial Function (A Total Function is also a
Partial Function.)
NOT a Partial Function
f (lt1,2gt, ß),(lt1,3gt, ß),(lt1,3gt, ?) lt1,3gt not
unique
49
Special Functions

• Identity Function
• IA A ? A
• IA (x, x) x ? A
• Constant Function
• C A ? B
• C (x, c) x ? A ? c ? B
• Often A and B are the same
• C A ? A
• C (x, c) x ? A ? c ? A

50
Overloading

• The same name can be used for different functions
  depending on the domain.
• Examples
• a b
• Means number subtraction if a and b are numbers
• Means set subtraction if a and b are sets
• a b
• Implemented differently for integers and reals
• Could potentially be overridden and implemented
  by a programmer for very long integers

51
Composition of Functions
g
1
a
a
f
2
ß
b
3
4
c

• Composition is written
• Range space of f domain space of g

f(a) 2 g(2) a g(f(a)) a
gf(a) a
f(b) 2 g(2) a g(f(b)) a
gf(b) a
f(c) 4 g(4) ß g(f(c)) ß
gf(c) ß
52
Injection

• Injection one-to-one or 1-1
• ?x?y(f(x) f(y) ? x y)
• For f A ? B, the elements in B are hit at
  most once

Injective
NOT Injective
a
1
a
1
b
b
2
2
c
c
d
3
d
3
y
y
x
x
53
Surjection

• Surjection onto
• ?y?x(y f(x))
• For f A ? B, the elements in B are all hit at
  least once

Surjective
NOT Surjective
a
a
1
1
2
2
b
b
3
3
c
c
4
4
y
y

not hit
x
x
54
Bijection

• Bijection one-to-one and onto or 1-1
  correspondence
• ?x?y(f(x) f(y) ? x y) ? ?y?x(y f(x))
• For f A ? B, every B element is hit once and
  only once

NOT Surjective NOT injective
Bijective
NOT Bijective
a
a
1
1
2
2
b
b
3
3
c
c
4
y
y
x
x
55
Notes on Bijection

• A B
• An extra B cannot be hit (not a surjection)
• An extra A requires that at least one B must be
  hit twice (not an injection)
• If f is a bijection, swapping the elements of the
  ordered pairs is a function
• Called the inverse
• Denoted f-1
• Is also a bijection
• f-1(f(x)) is the identity function, i.e.
  f-1(f(x)) x.

56
Notes on Bijection (contd)

• The inverse of an injection is a partial
  function.
• If f A ?B is an injection, then f-1 is a
  partial function

f
f-1
a
1
1
a
b
b
2
2
c
c
d
3
3
d

• Restricting the range space of an injective
  function to the range yields a bijection
• Remove b

57
Which is bigger? N or R0..1?
Assume N R0..1, then there exists a
bijection 1 0.34234 2 0.34987 diagonalizati
on 3 0.00040 But now there exists a number in
R0..1 such that d1 not 3, d2 not 4, d3
not 0, . Hence, not surjective and thus not
bijective.
58
Which is bigger? N or Z?
Since g f-1, there is a bijection from N to Z
and thus N Z. Countable if same
cardinality as some subset of N. Alternatively, a
set S is countable if there exists an injective
function from S to N.