which make them
particularly useful
in everyday life.
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.
With these three
points in mind,
the question
is: ,
How can we use
real numbers in
practical
calculations?
What rules
apply?
In addition, the
following
three math
properties of
equivalence
determine when one
algebraic quantity
can be substituted
for another without
changing the
original value.
Properties of
Equivalence
 click
description
Return
To
"Top
of
Page"
click
here
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.