Everyone is
familiar with this
idea since all
measurements
(weight, the
purchasing power of
money, the speed of
a car, etc.) depend
upon the fact that
some numbers have a
higher value than
other numbers. Ten
is greater than
five, and five is
greater than four .
. . and so on. This
is an important math
property.
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Examples
of
Reflexive,
Symmetric,
and
Transitive
Equivalence
Properties
An Equivalence
Relationship
always
satisfies three
conditions:
-
Reflexive
Property
-
Symmetric
Property
-
Transitive
Property
Is
the
"="
(the equal
sign)
an
equivalence
relation
for all
real
numbers?
a =
any real
number,
b =
any real
number,
c =
any real
number
Reflexive
Property
test:
Does
a =
a
for all real
numbers?
True
- This
statement is
true for all
real
numbers.
For example: 3
= 3
Symmetric
Property
test:
Does
a =
b
and
b =
a
hold true for
all real
numbers?
True
- These two
statements are
true for all
real
numbers.
For example: 3
= 4 - 1 and 4 -
1 = 3 are both
true.
Transitive
Property
test:
Does
a =
b
and
b =
c
(imply)
a
= c
?
True
- These
statements are
true for all
real
numbers.
For example:
3 = 4 - 1 and 4
- 1 = 5 - 2
(implies)
3 = 5 -
2.
True:
all three
property
tests are
true
.
The
"="
(equal
sign)
is
an
equivalence
relation
for all
real
numbers.
This means
that the
values
on either
side of
the
"="
(equal
sign)
can
be
substituted
for one
another.
Is
the
">"
(the
greater
than
symbol)
an
equivalence
relation
for all
real
numbers?
a =
any real
number,
b =
any real
number,
c =
any real
number
Reflexive
Property
test:
Is a
> a
for all real
numbers?
False
- This
statement is
false for all
real
numbers.
For example: 3
> 3 - this
statement
is not
true.
Symmetric
Property
test:
Does
a
> b
(imply)
b
>
a
is true
for all
real
numbers?
False
- This
statement is
false for all
real
numbers.
For example:
3 > 2
(implies)
2 > 3
is
not
true.
Transitive
Property
test:
Does
a
> b
and
b
> c
(imply)
a
>
c
?
True
- These
statements are
true for all
real
numbers.
For example:
3 > 2 and 2
> 1
(implies)
3 >
1.
False:
two of
the three
property tests
are
false
.
The
">"
(greater
than
symbol) is
not
an
equivalence
relation
for all
real
numbers.
This means
that the
values
on either
side of
the
">"
(greater
than
symbol)
cannot
be
substituted
for one
another.
Is
the
"
"
(the
greater
than or
equal
to
symbol) an
equivalence
relation
for all
real
numbers?
a =
any real
number,
b =
any real
number,
c =
any real
number
Reflexive
Property
test:
Is a
a
for all
real
numbers?
True
- This
statement is
true for all
real
numbers.
For example: 3
3 -
this
statement
is
true.
Symmetric
Property
test:
Does
a
b
(imply)
b
a
is true
for all
real
numbers?
False
- This
statement is
false for all
real
numbers.
For example:
3
2
(implies)
2
3
is
not
true.
Transitive
Property
test:
Does
a
b
and
b
c
(imply)
a
c
?
True
- These
statements are
true for all
real
numbers.
For example:
3
2
and 2
1
(implies)
3
1.
False:
one of
the three
property tests
is false
.
The "

"
(greater
than or
equal to
symbol) is
not
an
equivalence
relation
for all
real
numbers.
This means
that the
values
on either
side of
the "

"
(greater
than or
equal to
symbol)
cannot
be
substituted
for one
another.