Log i cal Rela tions
1. ∨
⇒
x P x
P c
( )
( ), where c is an arbitrary element of universe.
2. P c
P x
x
( )
( ),
⇒ ℑ
where c is an arbitrary element of universe.
3. ∨ ¬
⇔ ¬ ℑ
x
x
P x
P x
( )
( )
4. ∨
⇔ ¬
x P x
P x
( )
( )
5. ℑ ¬
⇔ ¬ ∨
x
x
P x
P x
( )
( )
6. [
( )
]
[ ( )
]
∨
∧ ⇔ ∨
∧
x
x
P x Q
P x Q
7. [
( )
]
[ ( )
]
∨
∨ ⇔ ∨
∨
x
x
P x Q
P x Q
8. [
( )
( )]
[ ( )
( )]
∨
∧ ∨
⇔ ∨
∧
x
x
x
P x
Q x
P x Q x
9. [
( )
( )]
[ ( )
( )]
∨
∨ ∨
⇔ ∨
∨
x
x
x
P x
Q x
P x Q x
10. [
( )
]
[ ( )
]
ℑ
∧ ⇔ ℑ
∧
x
x
P x Q
P x Q
11. [
( )
]
[ ( )
]
ℑ
∨ ⇔ ℑ
∨
x
x
P x Q
P x Q
12. ℑ
∧
⇔ ℑ
∧ ℑ
x
x
x
P x Q x
P x
Q x
[ ( )
( )] [
( )
( )]
13. [
( )
( )] [
( )
( )]
ℑ
∨ ℑ
⇔ ℑ
∨
x
x
x
P x
Q x
P x Q x
Ì Exam ple 8.4.1: Let P(x, y, z) denote xy = z; E(x, y) denote x = y; and
G(x, y) denote x > y.
Let the universe of discourse be the integers. Transcribe the following into
logical notation.
(a) If y = 1, then xy = x for any x.
(b) If xy ≠ 0, then x ≠ 0 and y ≠ 0
(c) If xy = 0, then x = 0 or y = 0.
(d) 3x = 6 if and only if x = 2.
(e) There is no solution to x
2 = y unless y ≥ 0.
(f) x < z is a necessary condition for x < y and y < z.
(g) x y
≤ and y x
≤ is a sufficient condition for y = x.
(h) If x < y and z < 0, then xz < yz.
(i) It cannot happen that x = y and x < y.
(j) If x < y then for some z such that z < 0, xz > yz.
(k) There is an x such that for every y and z, xy = xz.
Solu tion
(a) If y = 1, then xy = x for any x.
∨
⇒ ∨
y
x
E y
P x y x
[ ( , )
[ ( , , )]]
1
(b) If xy ≠ 0, then x ≠ 0 and y ≠ 0.
∨ ∨ ¬
⇒ ¬
∧ ¬
x
y
P x y
E y
E x
[
( , , )
( , )
( , )]
0
0
0
(c) If xy = 0, then x = 0, and y = 0.
∨ ∨
⇒
∧
x
y P x y
E y E x
[ ( , , )
( , )
( , )]
0
0
0
Prop o si tions and Pred i cates
283