The Symmetric Property of Relations
The symmetric property is defined by a conditional statement.
Let ∼ be a relational symbol.
If x ∼ y implies y ∼ x for every x, y ∈ U, then ∼ is symmetric.
Definition 1 (Symmetric Relation). Let ∼ be a relation on set U. The relation
∼ is symmetric if x ∼ y implies y ∼ x for all x, y ∈ U.
In symbols, ∼ is symmetric when
(∀ x, y ∈ U)(( x ∼ y) → (y ∼ x ))
Representations of Symmetric Relations
If you recall, a relation is a set ∼ of ordered pairs. Symmetric says
if ( x, y) ∈ ∼, then (y, x ) ∈ ∼.
If you recall, a relation is an adjacency matrix, Symmetric says the
value in row x, column y equals the value in row y, column x.
a
a
b
c
d
..
.
u
b
c
··· u
d
x • • • ··· •
• x • 0 ··· 0
• • x 1 ··· 1
• 0 1 x ··· 0
.. .. .. ..
.
. . . . · · · ..
• 0 1 0 ··· x
The diagonal values x do not matter.
Representations of Symmetric Relations
If you know about matrices, a relation is symmetric when its
adjacency matrix equals its transpose. Row k of A is column k of A T .
a
A=b
c
d
a
A=b
c
d
a
b
c
d
0
1
1
0
1
1
0
0
1
0
0
1
0
T
0=A
1
1
a
b
c
d
0
1
0
0
1
1
1
0
1
0
0
0
0
T
0 6= A
1
1
1
is symmetric
is not symmetric
Examples of Symmetric Relations
The following relations are symmetric.
• Equality on the set of integers. (∀ a, b ∈ Z)(( a = b) → (b = a)).
• Not equal on the set of integers. (∀ a, b ∈ Z)(( a 6= b) → (b 6= a)).
• Congruence mod n on the set of integers. (∀ a ∈ Z)(( a ≡ b (mod n)) →
(b ≡ a (mod n))).
If a − b is a multiple of n, then b − a is a multiple of n
Examples of Symmetric Relations
The following relations are symmetric.
• Points on an origin-centered circle: (∀( x, y) ∈ R2 )(( x2 + y2 − 1 = 0) →
(y2 + x2 − 1 = 0)).
y
•
x
•
• Even products on the integers:
If xy is even, then yx is even.
Examples of Relations that are Not Symmetric
The following relations are not symmetric.
• Less than on the set of integers. A counterexample is, 1 < 2
but 2 6< 1.
• Divides on the set of natural numbers. A counterexample is,
1 | 2 but 2 - 1.
• Proper subset on 2N , the power set N. A counterexample is,
{0} ⊂ {0, 1} but {0, 1} 6⊂ {0}.
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Counting Symmetric Relations
It is interesting to count symmetric relations on a finite set.
Theorem 2 (The Number of Symmetric Relations). Let U be a set with cardinality
n. There are
p
2n(n+1)/2 = 2n(n+1)
symmetric relations on U.
Recall, a relation on U can be represented as an n × n adjacency
matrix A.
There are n × n = n2 entries in A, and each entries can take on
one of 2 values.
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There are 2n different adjacency matrices (relations) that can be
defined.
Using Adjacency Matrices to Count Symmetric Relations
Continuing from the previous slide.
To be symmetric, the matrix A must have equal values in symmetric entries: the value in row x, column y equals the value in row y,
column x.
Recall, there are
1 + 2 + 3 + · · · + ( n − 1) + n =
n ( n + 1)
2
entries on or below the main diagonal of A.
The value in an entry of A can either be True or False.
But, once a value for an off-diagonal entry is chosen the value
of the symmetric entry is also determined.
There are
2n(n+1)/2 =
p
2n ( n +1)
ways to choose values on or below the main diagonal of A, and for
each of these choice you can construct a symmetric matrix.
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Counting Symmetric Relations
Here’s another way to count symmetric relations. Let U be a set
with cardinality n.
Let name and order the elements of U.
U = { u 0 , u 1 , u 2 , . . . , u n −2 , u n −1 }
A relation on U can be represented as a subset ∼ of the Cartesian
product U × U.
There are n × n = n2 ordered pairs in U × U.
To be symmetric, if ( x, y) ∈ ∼ then (y, x ) ∈ ∼.
Consider p( j, k) = u j ∼ uk as a predicate statement. When creating a relation you can set p( j, k) to either True or False.
Counting Symmetric Relations
To make a symmetric relation you choose to set
u j ∼ uk
to True or False
for j = 0, 1, . . . , n − 1 and k = j, j + 1, . . . , n − 1. For a fixed j, the index
k steps through n − j decisions.
( u0 , u0 ) ∈ ∼ ( u0 , u1 ) ∈ ∼ ( u0 , u2 ) ∈ ∼ · · ·
( u1 , u1 ) ∈ ∼ ( u1 , u2 ) ∈ ∼ · · ·
( u2 , u2 ) ∈ ∼ · · ·
..
.
( u 0 , u n −1 ) ∈ ∼
( u 1 , u n −1 ) ∈ ∼
( u 2 , u n −1 ) ∈ ∼
..
.
( u n −1 , u n −1 ) ∈ ∼
There are 2n(n+1)/2 symmetric adjacency matrices.
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Related Relation
• A relation ∼ is not symmetric when there are values x and y
such that x ∼ y and y 6∼ x. In symbols,
(∃ x, y ∈ U)(( x ∼ y) ∧ (y 6∼ x ))
• A relation ∼ is asymmetric when ( x ∼ y) ∧ (y ∼ x ), for all x and
y. In symbols,
(∀ x, y ∈ U)(( x ∼ y) → (y 6∼ x ))
• A relation ∼ is antisymmetric when for all x and y, if x 6= y, then
x 6∼ y or y 6∼ x. In symbols,
(∀ x, y ∈ U)(( x 6= y) → (( x 6∼ y) ∨ (y 6∼ x ))
Problems on Symmetric Relations
Show your understanding of this topic by completing the problems found at Symmetric Relations.
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