Fill in the blanks in the following proof that for all integers a and b, if a | b then a | ( − b ) . Proof: Suppose a and b are any integers such that __(a)___ By definition of divisibility, there exists an integer r such that __(b)__ By substitution, − b = − ( a r ) = a ( − r ) .Let t = ____(c)____ Then t is an integer because t = ( − 1 ) ⋅ r , and both -1 and r are integers. Thus, by substitution, − b = a t , where t is an integer, and so by definition of divisibility __(d)___ as was to he shown.
Fill in the blanks in the following proof that for all integers a and b, if a | b then a | ( − b ) . Proof: Suppose a and b are any integers such that __(a)___ By definition of divisibility, there exists an integer r such that __(b)__ By substitution, − b = − ( a r ) = a ( − r ) .Let t = ____(c)____ Then t is an integer because t = ( − 1 ) ⋅ r , and both -1 and r are integers. Thus, by substitution, − b = a t , where t is an integer, and so by definition of divisibility __(d)___ as was to he shown.
Solution Summary: The author explains how to fill the blanks in the given proof. Suppose a and b are any integers such that underset_a|b
Fill in the blanks in the following proof that for all integers a and b, if
a
|
b
then
a
|
(
−
b
)
. Proof: Suppose a and b are any integers such that __(a)___ By definition of divisibility, there exists an integer r such that __(b)__ By substitution,
−
b
=
−
(
a
r
)
=
a
(
−
r
)
.Let
t
=
____(c)____ Then t is an integer because
t
=
(
−
1
)
⋅
r
, and both -1 and r are integers. Thus, by substitution,
−
b
=
a
t
, where t is an integer, and so by definition of divisibility __(d)___ as was to he shown.
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY