McCombs Math 381
Contrapositive and Contradiction
Proof by Contrapositive:
To prove “if A, then B”:
Assume
not
B, and then show
not
A.
Proof by Contradiction:
To prove “if A, then B”:
Assume A and
not
B, and then show
you get a
contradiction
.
Prove each of the following by the
contrapositive
method.
1. If
x
and
y
are two integers whose product is even, then at least one of the two must
be even.
:
A
x y
!
=
even
:
A
¬
x y
!
=
odd
:
B
x
=
even or
y
=
even
:
B
¬
x
=
odd and
y
=
odd
We need to show that
B
A
¬
!
¬
.
x
=
odd and
y
=
odd means that
2
1
x
k
=
+
and
2
1
y
m
=
+
, for some integers
k
and
m
.
Thus,
(
)(
)
(
)
2
1
2
1
4
2
2
1
2 2
1
x y
k
m
km
k
m
km
k
m
!
=
+
+
=
+
+
+
=
+
+
+
.
Therefore,
x y
!
=
odd, by the definition of an odd integer.
Hence, we have shown that
B
A
¬
!
¬
.
2. If
x
and
y
are two integers whose product is odd, then both must be odd.
:
A
x y
!
=
odd
:
A
¬
x y
!
=
even
:
B
x
=
odd and
y
=
odd
:
B
¬
x
=
even or
y
=
even
We need to show that
B
A
¬
!
¬
.
x
=
even or
y
=
even means that
2
x
k
=
or
2
y
m
=
, for some integers
k
and
m
.
If
2
x
k
=
, then
(
)
(
)
2
2
x y
k
y
ky
!
=
=
. Thus,
x y
!
=
even, by the definition of an
even integer.
Similarly, if
2
y
m
=
, then
(
)
(
)
2
2
x y
x
m
xm
!
=
=
. Thus,
x y
!
=
even, by the
definition of an even integer.
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 Summer '11
 NA
 Math, Contrapositive, 4k, 2Km

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