(c) ℑ ∨
⋅ =
y
x x y
[
]
1
(d) ℑ ∨
⋅ =
y
x x y x
[
]
Solu tion
(a) ∨ ℑ
⋅ =
y
y x y
[
]
0
y = 0 makes it TRUE.
(b) ∨ ℑ
⋅ =
x
y x y
! [
]
1 False
(c) ℑ ∨
⋅ =
⋅ =
⋅ =
y
x x y
y
y
[
]
1
1
1
2
1
y
I
= ∉
1
2
.
False.
(d) ℑ ∨
⋅ =
⋅ =
⋅ =
=
y
x x y x
y
[
]
.
1 1 1
2 1 2
1 makes it true
Ì Exam ple 8.3.4: Let the universe be the integers. For each of the
following assertions, find a predicate P which makes the implication false.
(a) ∨ ℑ
⇒ ℑ ∨
x
x
y P x y
y
P x y
!
( , )
!
( , )
(b) ℑ ∨
⇒ ∨ ℑ
!
( , )
!
( , )
y
P x y
y P x y
x
x
Solu tion
(a) ∨ ℑ
⇒ ℑ ∨
x
x
y P x y
y
P x y
!
( , )
!
( , )
x + y = 0
LHS:
x
x
x
x
+ − =
=
+ − =
=
( )
( )
1 0
1
2 0
2
RHS:
ℑ
=
+ =
= −
! ,
,
y x
y
y
1
1
0
1
ℑ
=
+ =
= −
! ,
,
y x
y
y
2 2
0
2
P x y
x y
( , ) (
) .
is + = 0
Prop o si tions and Pred i cates
279