The notion of a relation

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Todays topics

• The notion of a relation
• properties of relations on a set

Relations

• A relation is a fundamental mathematical
  notion expressing
• a relationship between sets
• Its an abstract notion useful for modeling
  many different
• relationships

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Example. Let S be set of UCF students, and C be a
set of classes. Then we can consider the
relation is taking class from S to C
C
S
x
y
?
?
?
?
?
?
?
This relation can be described by the set of
pairs is taking class(x, y) x?S, y?C and
student x is taking class y
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More examples of relations

• parent-of
• child-of
• likes
• meet one another today
• less then (a, b) a, b ?A and altb
• where A1, 2, 10
• equal (a, b) a, b ?Power(A) and a b
  
• subset (a, b) a, b ?Power(A) and a ? b
• If R is set of real numbers, R ?R is set of
  points (x, y) in plane.
• circle(x, y) x, y ?R and x2y21

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• You dont need to give a meaningful name to a
  relation.
• The only thing that really matters about
  relations is that
• we know which elements in A are related to which
  element of B.
• A relation R is completely described if we know
  R-related pairs
• Suppose A 1, 2, 3, B r, s and we know
  1Rr, 2Rs, 3Rr,
• then we know everything we need to know about R.
• R can be specified by the list of pairs
• R (1, r), (2, s), (3, r)

• The Cartesian product contains all possible
  pairs
• A?B (1, r), (1, s), (2, r), (2, s), (3, r),
  (3, s)

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Definition. A binary relation from A to B is a
subset R ? A?B. Conversely, any subset of A?B
can be considered as a relation.
A binary relation on a set A is a relation from A
to A, R ? A?A.

We can also define a ternary relation on A as a
subset R ? A?A ?A and, in general, an n-ary
relation as a subset R ? A1?A2? ? An, if the
sets Ai are different.
How many binary relations from A to B are there?
A?B A?B
2A?B subsets ( relations)
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• Notation
• R ? A ? B a binary relation from a set A to a
  set B
• R (a, b) a ?A, b?B and a is related to b in
  some way
• (a, b)? R ? aRb ? a is related to b by R
• (a, b)? R ? aRb ? a is not related to b by R
• R ? A ? A is a relation on A

/
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• If R ? A ? B is a relation from A to B, there are
  two
• important sets associated with it
• Domain of R is subset of elements in A which
  are
• related to some element in B.
• For relation F (father) from A (set of living
  males)
• to B (living females) the domain is the set of
  living males,
• who have living daughters.
• Range of R is the set of elements in B for
  which
• are second elements of pairs in R.
• For the relation F the range is the set of
  living females
• who have living father.

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A relation can be represented as a matrix.
Let R?A ?B, A m, B n, then we can define
the matrix mik with m rows and n columns
Let A 1, 2, 3, B r, s, R (1, r), (2,
s), (3, r) ? A ?B Then the following matrix
represents this relation
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Consider the relation on A 1, 2, 3, 4, 5
represented by the following directed graph
R ? A ? A
All edges of this graph correspond to related
pairs
R (1,1), (1, 2), (2, 2), (2, 3), (3, 3), (4,
4), (4, 5), (5, 5)
What kind of relation is it?
R (x, y) x, y ? A and (xy) ? (x1y)
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Properties of relations R ? A ?A
Definition. Let R ? A ?A be a binary relation on
A. 1) R is reflexive if for all a ?A , aRa, i.
e. (a, a) ?R. (In words, each element of A is
related to itself via R) 2) R is irreflexive if
for all a ?A, (a, a)?R. (In words, none of the
elements in A is related to itself via R) 3) R
is symmetric if aRb implies bRa, i. e. for all
(a, b) ?R, (b, a) ?R. (In words, whenever a
is related to b, b is related to a). 4) R
is anti-symmetric if aRb and bRa imply ab.
Equivalently, this means if a?b, then (a, b)?R
implies (b, a)?R. (In words, whenever a is
related to b and a?b, then b is not related to a).
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Symmetric, anti-symmetric or none?
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5) R is transitive if aRb and bRc imply aRc, i.
e. if (a, b)?R and (b, c) ?R, then (a, c) ?R.
(In words, whenever a is related to b and b is
related to c, then a is related to c).
Which graphs represent transitive relations on
a, b, c ?
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Give example of a relation on A a, b, c that
is
a) reflexive but not symmetric
R(a, a), (b, b), (c, c), (a, b)
b) symmetric but not reflexive and not transitive
R(a, b), (b, a), (b, b), (b, c), (c, b)
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Counting relations.
Consider a finite set A with An.

• How many reflexive relations are there on A?

There exist n2 pairs on A, n of them are
self-loops, like (a, a). A reflexive relation
must include all n self-loops. So, we are left
with 2 choices (include or not) for any of n2 ?n
pairs (a, b), of distinct elements, a?b. So
the total number of different reflexive relations
is 2n?n?n.

• The next question is how many irreflexive
  relations
• are there on A? (This is for you to figure out).

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• How many symmetric relations on A ?

A n
n (n2?n)/2 n?(n1)/2
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• How many anti-symmetric relations are on A?

We can include any number of self-loops. For any
couple of distinct elements a?b we have now 3
choices include (a, b) but not include (b,
a) include (b, a), but not include (a,
b) include neither (a, b) nor (b, a). We have
n?(n ?1)/2 couples and it gives 3n(n?1)/2
possible selections. This must be multiplied by
2n possible subsets of self-loops. The total
number of anti-symmetric relations is 2n?
3n(n?1)/2
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Suppose R1 and R2 are two relations on A.
Lets prove or disprove the following
statements.
a) R1 and R2 are reflexive ? R1? R2 is
reflexive?
Proof. Assume that R1 and R2 are reflexive. Take
any a ?A, We need to show that (a, a) ? R1? R2.
. But since both R1 and R2 are reflexive, (a,
a)?R1 and (a, a)?R2, that implies (a, a) ? R1? R2
b) R1 and R2 are symmetric ? R1? R2 is
symmetric?
Proof. Assume that R1 and R2 are symmetric. Then
take any pair (x, y) ? R1? R2 to show that (y,
x) ? R1? R2. (x, y) ? R1? R2 implies that (x, y)
? R1 and (x, y) ? R2,, but since both R1 and R2
are symmetric, it implies that (y, x)?R1 and (y,
x)?R, i. e. (y, x)?R1? R2. QED
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Some more statements you can think about
c) R1 and R2 are anti-symmetric ? R1? R2 is
anti-symmetric?
d) R1 and R2 are transitive ? R1? R2 is
transitive?
f) R1 and R2 are reflexive ? R1 ? R2 is
reflexive?
g) R1 and R2 are symmetric ? R1 ? R2 is
symmetric?
h) R1 and R2 are anti-symmetric ? R1 ? R2 is
anti-symmetric?