For any x, x belongs to set a →
2; Symmetry: Mathematically, if any.
a
and
b
belong to
X, the following statement is still valid, and then set.
X
Binary relation opening
rare
Is symmetrical: "If
a
involve
B, then
b
involve
Answer. "
Mathematically expressed as:
& lt Mathematics & gt\forall
One,
b
\ in
x，\
a
rare
b
\ Right Arrow
\;
b
rare
a & lt/math & gt；
For example, "marriage" is a symmetrical relationship; "Less than" is not symmetrical.
Symmetric relation is not antisymmetric relation (aRb
and
brassiere
get
b
=
The antonym of a). Some relationships are both symmetric and anti-symmetric, such as "equality"; Some relationships are neither symmetric nor anti-symmetric, such as "divisibility" of integers; Some relationships are symmetric but not anti-symmetric, such as "module"
n
Congruence "; Some relationships are asymmetric but anti-symmetric, such as "less than".
3 Transmission: In logic and mathematics, if all
A, B and C
belong to
X, the following statement is still valid, and then set.
X
Binary relation opening
rare
Transmitted: "If A"
involve
b
and
b
involve
c，
rule
a
involve
C. "
Mathematically expressed as:
& lt Mathematics & gt\forall
One,
b，
c
\ in
x，\
a
rare
b
\ and
b
rare
c
\;
\ Right Arrow
a
rare
c & lt/math & gt；
4 anti-reflexive:
5 antisymmetry: mathematically, if all
a
and
b
belong to
X, the following statement is still valid, and then set.
X
Binary relation opening
rare
Is anti-symmetric: "If all
a
and
b
belong to
X if
a
involve
b
and
b
involve
So, a
a
=
B. "
Mathematically expressed as:
& lt Mathematics & gt\forall
One,
b
\ in
x，\
a
rare
b
\ and
b
rare
a
\;
\ Right Arrow
\;
a
=
b & lt/math & gt；
Strict inequality is antisymmetric; in fact
a
& lt
b
and
b
& lt
a
It is impossible, so the anti-symmetry of strict inequality is an empty truth (vacuously
Really.
Note that anti-symmetric relation is not symmetric relation (aRb
get
Antisense of bRa). Some relationships are both symmetric and anti-symmetric, such as "equality"; Some relationships are neither symmetric nor anti-symmetric, such as "divisibility" of integers; Some relationships are symmetric but not anti-symmetric, such as "module"
n
Congruence "; Some relationships are asymmetric but anti-symmetric, such as "less than".
An antisymmetric relation satisfying transitivity and reflexivity is called a partial order relation.