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: M. , -.
8 6 E
M
M
5, @ I 0
= 3. 8 0
6 0 - < :8
/4 :I @
: 4 5: B
J ,O 4
F 3
/ i. 6 0 - :. 0 4
6 / E
8
S<8 6 9 -.
8 6,
=6 @ M
G 9 :F6 /3 /6 8
8
68 0 /63 0
M ,4
:, [4 3 0 4
6
F6 7
1 9 /3 0.
]
5: B :, 4
:.
3
:. 0 4 /6
, .
/ 0 - :38
5.
(y))
0010
,M
@6
f0000
8,
E
/
8I
,
:8 0
6;
(x, y),
(x,
(y)),
(y,
(x)),
(
(x, y)).
5z
,0
6 0.
true
34
-
,
EW
64
J 8M
: i4
89
G @E 9
6
:F6
/6
68 0
1
~
~
ˆ ‡‹
bool
89
0
:8 -
} {
(0, 0, 0, 1)
unsigned int
bool
,
J
{
J
76
5: B 0
3 80
:. 0 4 :3. S/6 f
J, , 3
4
, 0 4 6,
,.
B 8g
6 0- , A
,
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5
F6 9 0.
5: B ,
E -.
3
:. 0 4 0
8 0U 6 0L
Ul 6,
/68 , 0
L
Ul
<3.
=6 9 U
60 k
Z
/. B G
Xk
6 06, U
k
.0 G Z
B :8
: I4
/ 6M
@ 8 7:6
\
.. 6 0
69 :8 6,
0
{
?>
:D
f1100 (x, y) =
f0111 (x, y) =
J
4
B
(x,
=
=
=
=
30
6 0E0
.6
6
<6 9
:
,0
:8
.
?
G \
KJ@ 9
..
6
/ i.
:8 9
6
/,68
S<8
/6
=6 9
/=6
G
6
M :,
& 6 0
M
0 @I
8 0- 6 0/68
5: B
8, M
,. <68
34 :
68 0 0
E M 8 0@ 9
6
04 =
6 0- J
9
/6 0 4 0 D
,
89 7
, 586 9
/
:
:8 86
F
M :3.
,4
G ,, 0
=
,6
/6
f0001 (x, y)
f0010 (x, y)
f0100 (x, y)
f1000 (x, y)
G
b
6
:M
/: 05 / i. 6 T
< ,
,6 4 8 9 /8
9 @ 9 78
,, G
0 4
F:6 6 - 86 I9
6/ 0@
8 9 =6 . B
0
0@
:8 =6
4
:,8
:4 :
6 8
/F 0 : M
4
89 4
0@ 0 4
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G
6P 6M ,
4
3 :T 0
:. 0 4 6 0.
M
0. J 6I
Y 69
,,
:F6
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:4
Proof.
f
(f(0, 0), f(0, 1), f(1, 0), f(1, 1))
0001
f b1 b2 b3 b4
b1 b2 b3 b4
= f0001
||
&&
< >
<=
==
>=
bool
!=
<
==
=
7 + x < y && y != 3 * z
((7 + x ) < y ) && ( y != (3 * z )).
a == b == c
2.3.2 The type bool
a == b && b == c.
o
og
ty
x
w
v
u
t
s
rq
~
~
‹
ˆ
~
~
ˆ~
‹
‹
x
0
.
-.
76
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M
=
M
M
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0
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89
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6 Z-
B
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6 99
38
@ 09
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89
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Xk
5: B
3
:. 0 4
{
6 Z-
Naming.
‹
3.
J,
/8 0.
=/6
B
/8 0.
/=6
.
89
89
:. 0 4
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5,
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6 0-
0
8 00
5,
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M
6I
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M
78
59
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8,6
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M
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5: 4
,
6 E8
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6 @8
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78
J 9M4
DB
60
/-
G
x==0
‹
bool
04
/. B
6 0/8
04
<6 !
&
"

+ /F6
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89
6
,
/,.
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69 /
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69
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3
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0
9.
o
gp
gp
gg
gg
gg
gg
go
g
0
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0
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S=/ 6
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8 F
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0 8-8 / 0
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6 6
M M
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8
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:.
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K
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l 9
56
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0. ,
4
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=
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3-
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86 I
9
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,.
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8
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8
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=
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6
M 66 0
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4
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8
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0
=
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# ,,
:
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M 8
0
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4
76
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HG, . 0
/
6
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6 0 - /= 6 80 8
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/ 0. T
J 8 99 @ M
5,
8
6M
90E 6 0 - 7 9
Z :6 T
-4
6 9 586 9
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M
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7
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8 60
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7
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6
6 & 758 9 /6 M
0
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0
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:
A 6
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E . 9B
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6
:
8
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B
M
. :. . . 0
:6 0 =/6 66 < /=6
:
8
/8 0. 0. M /6 00 8
6 0M
6
6 0- 7
6 9 58 9
B 68
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6/= 6 0
:8 / 4
M6 F 78 . 0
5 9 /=6
68 0 :8
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0.
8
0
J 8 99
6 Z:8
8 9.
5F.
,
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05
8
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69 69 69 69 69 69 69 69 /
4
0B 0B 0B 0B 0B 0B 0B 0B F 0
Prec. Assoc.
6 0-
&&
7 8 :8
6
56 9
M 7
.
189
B
/68
G
/8 0.
:8 J
J
,. M J
:8 B
.
/=6
/=6
.
3 -4 J
6 0- J
:
E4
/6 M
/.
F
F/8 G
: M4
<4
/=.
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0
M4d
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:
6
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8,
Q5
4
768
:9
@ 09
6
EI
/
4
:6 00
6
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0.
6
6 Z-
6 0 - 758 9
:F6 :8. 0 4
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.
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56 S/
9 =6
, ,
8, :,. 4
M4 :,
5,3 4
,6, 7. 9
7:
: M4 4
F
6 P 9.
3 F4
:. 0 4 38 9
:
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9
:8 g 8 0
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M 8
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/=6
8
E N. / 0.
6 ,
/
7/6 =.
663
/66 0 - M
,
4 83
:., /3.
6 1M
F
:4
0@
<
,6
6 Z-
86 9
8
$
<
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M
0.
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5
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.
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5
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B
,
:,. 4
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S/
:8 W
6
=6
G 9 *
"
S<8
6
/ i.
4
768
:9
0
/. B
J<
Q5
: 4Y
0.
8
[
<, 4
=/6 9
6]
=6
:8
M
6Z
:8
M
&
0.
60
:3.
7/6
,4
:., 4
GH,
60
: -4
6
/F 0
89
.
/=6
:8
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-
F.6
7/6
:,. 4
:3.
6Z
76 4
3
:F8 7/6
,
hgp W E8
04
_ ogh 0 6J gY 6 - Z
6
6 =9 EEN .B 6
9
8, U 1 : 9
k A
8 G Z E.
:=. 4 /68 :
6 : /.
B
/6
: 4 <86 <5
@<, M 68 9
8
/6 0 B . B
I.
9 0 E -.
34 68 / \ 0.
F/6 9 4 6
I , 0 4 S/
:88 < - =6
,
M 86 0 ,
9. - L
F < Ul
J 3, 4 8 : 4
04
6
L9 :38 4 / 0
<
,.
60 5 L ,
/ - F5 . B
/F. 4 , 05 Xk
:
5. ,6 l 8
,
M
/.
0
:8 M
:, 4
0.
68 A
@ 9. B 6 0
=9 .
86 6 0 F 4 .
:3 4 / - 38 9 /6 0 : 4 7 .6
58 9 /= /6 M4
04 6
8 94 4 /8 0. 3 0
4 ,
:. 4
8 0 :3. /,68
:. 4 6
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7/
:,8 6 0 :3. 9.
6
M M 7/ F 4
8, 0. 6 0 38 9
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5, <
3- 4
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F
,.
89
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7/6
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6
S<8
/.
6 Z- < B
6M 5
1 89
0
/. :. 4
F: .
1 8 < [B
/. B
< 80
5 668 9 <
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68 0 :38 9
0 4
8 0 - 53 M
: 4 W :. 0 4
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1 6
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:
F5 E. A
F86 :
Y 8
5M
6,
04
0.
@8 M
F.6
,
83 I
A
0.
J <4
0
int
8,
6
#
# :4
6 Q5
F 4 5Q
38
8 9 /86 0
:8 9 8 94 0@ 4 /.
0@
6
M
Q5
89
9.
Arity
4
F4
38
/. 9
9.
6
Operator
!
<
>
<=
>=
==
!=
&&
||
@8
E
<,8
6 0-
6 I94
0
8 06 0E0
.
.
/=6
:8
,. M
,,
=.
0D
,4
!
! ! !
# Description
1
4
/,6 0
68 0
:M
Conversion and promotion.
,
368
Z
,4
0
/8 0.
GH,
.
/=6
x >= y && x + 1 <= z && y > 5 * z && y > 7 * z
0
:
"
Table 2:
J =6
0@
-
M
/66 4d
768
!( x < y || x + 1 > z ) && !( y <= 5 * z || !( y > 7 * z ))
~
0
bool
:
-8 <8 6 Z0 6
89 4 U
.9 , L
8 Ul
/6 0 - /6 0 B
5: B
9. 6 0
3
F4
:. 0 4
38
9. <6 L
/
/=6 4 , 4
8
8 :38 ,. 9
:. 0 4 < :
A
, 8 E.
:38 6 0 - :
<
6I 8 8
6 04 ,
S/= :38 4 6
6,, :6 N / : M4 @
!
/6 0 J < P ,. 66 -d /.
B /
U W L ghh J Ul
_ g o
E mY J 6 0- 6 Z
69 ,, <3.
:F6 =.
/6 , 4
6
89 60
0@ S/
=6 =6
, 4 ,,
< 4 :. 4
= 9 6, 4
3 44 7
@ 0 9 58 9
:3. 68 0
7/6 :. M
60
J M 6 0:D <
/.
/=8 6
0 :F6
53 4 /6
/9
G 8 89
0@
6
G=
. 0.
/=6 E
:8 4
M 3/,68 6 0
.
:3. /=6
7/6 :8
60 M
.
M B
!( x && y ) == (! x || ! y )
"
0 0. 0.
-8 :0
G 60
. :8 /6 : 4 /. B J M =,6
<
L 36 0 4
7
8 0 /=.
:. 4 < 0. : 4
. J /6 0
F6 :. 0 4
F
0,
D 8
9. 4 0 B 0@ 9
J ,0 , 6- 6
8 : =
,
=6 6 0 4 .
3 4 / 6 0B
89 F :3. 8 9 .
0@ 6 0
7/6 =6 / :., 4 , 4 /.6
=
:8
/. B
E G J
M
-4 0 -4 /6 N
3 , 6
6 0 - :3. G
0
7/6 6 ?>
? :,. 4 78
56 9
:,8 0 , 4
&
/8 M :6 M M 6T 4
F5 M :,3.
/8 0.
:8
7/6
66 0 6 I 6
0
, 8 M
De Morgan’s laws.
bool
bool b = 5; // b is initialized to true
int i = b ; // i is initialized to 1
!( x || y ) == (! x && ! y ) .
2.3.3 Short circuit evaluation
||
&&
||
x != 0 && z / x > y
2.3.4 Details
om
o
ty
x
w
v
u
t
s
rq
:
B 5
<
b
32
L
{
,
Z
}
W
|
xn
g^ Y Y
7
Y
W
g^ Y
`
&
W
g^ Y
* `
→
W ^ Y &
|
Y
,4
M
/, 0 4
4
5I
06 4
7
`
b
34
80
64
,.
n
...
|
x2
&
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M
/, 0 4
4
5I
06 4
7
Xk
!
$
,4
8,
Xk
{
&
% c
f:
x1
&
Y
L
G
J 6 4W
J
|
#
b
0@
E 9.
6I
66, W
<8
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/=
G:Y
4
7.
97:
4
F.
6 Iz
30
J , ,.
B
:8 Y
/8 M
04
<6 0
HG3 4
9.
F4
38
G 9
:8
/6 M
89
:. 0 4
89
.
/=6
/8 0.
<, 4
=6 9
/6 M
: 4W ,
:0
53 9 8
: M4 M
8
F F6 4
7:
=6
3
04
bool
"
7
Ul
L
J
|
x, y, z
34
80
64
,.
Xk
Ul
8,
,4
G
J 6 4W
J
G
J 6 4W
J
Ul
L
U
,
,
J
(
G
J 6 4W
J
J
J
(
b
z)
% +c
#
#
(
!
"
%
(y
" !
z=x
&
(x ↑ y) ↑ z = x ↑ (y ↑ z)
‘
n
&
x 1 , . . . , xn n
xi 1 i n
(x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z)
y)
|
04
<6 -
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W
^
/6 m Y
8
:8 M
5: M
^
W
e,
0
GH3 4
9.
F4
38
G :9
8
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89
:. 0 4
8. 9
/=6
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:4
7.
7: 94
F/
8
04
<6 -
,
:,. 4
6Y
78
59
68 0
8
F6 4
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6
S/=
6
.
B
e
=6 9
60
5: B
3
:. 0 4
,
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<3.
: I4
/
@8
\
..
6
:8 9
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6,
0
8
F6 4
:
/.
-
^
6/6 0
7:
E
.
76
/=
Y
W
0\
5: B
3
:. 0 4
e,
..
6
:8 9
5
8 I.
,0
34
,0
68
<6 0
:
8, I
M
/=, 4
.
76
.
76
/.
/=
W
g^ Y
:D
M
9
9
86 I
6I
5 -.
,
@.
5
JJ 0.
J
G
0
53 4
/8 9
/=8
Operational.
b
&
,
/ -.
78
59
8
:. 0 4
Y
:5
6/ M
:,8 0
M
6 06
6B
B .
0
53 4
/8 9
6 0:3.
63
=0
.
/=8
:4
HG,
0 S/
/3 4 =6
53 :,,. 4
06 4
,
78 : 4
5 9 7.
8 9
:. 0 4 7: 4
J
F
9.
F4
38
:8 9
/6 M
89
:. 0 4
89
.
/=6
/8 0.
"
!
e
G
&
0@
7.
7: 94
F
4
8
:. 0 4
:,
:,. 4
:,8
.M
/=6
7/6
:3.
6 0-
M
78
56 9
/
:8
6
GF
:8
4
,0
=6
6 0-
:A
E.
bool
%
(x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z)
#
<
6 Z/6 0
, 4 " ,0 8 0 9 ,. 0@ B "
=6 8,6 +
:,8 Q56 : . M 36 /6
/=6 . /6 B
8 6B ,
:. 0 4 M 0.
3
, /8 9 8
M4 8 0 /=
5,3 :. 4 /F.
6,, :8 <8
: M4 M6 0
S/ 6 0 - =6 8 0
/=6 ,, :3.
:. 4 ,
63
,4
: M4 , 0 ,. 0
68
F6 <6 0 8 B
, : <
3
:. 0 4 J , 0 :8 4
/,68 8 U 5: B
/ 05 3
0
5
6, 8 9 :. 4
J M G @9 E
:.
@ 9 3 -4
:
<86 M :3.
5: B /: 05
:4
G
G
%
Y
!
+ !
+ # +
!

<
:8
\M
=0
63
:3.
6 0-
M
/6 M
:,8 0
5: B
3
:. 0 4
:,8
5: M
69
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9.
J
=:.
4
G 0
@.
5
,
5 -.
JJ 9M
0
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34
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6 08, I
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:6 0
6
,
e,
gY
B :A
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.
,
Q5:6
86
8,
J
J
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4
0
8: 04
60
/F6
,
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6
/6 I
/=6
:6,
60
:M
h
6P
3
:. 0 4
:4
:
66,
768
8 0- 6
;
HG, 4
0
G 80
<
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+ Exercise 21
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Exercise 19
k
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Exercise 20
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Exercise 18
U
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6 0-
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i
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ϕf (x) = bi=0 f(ai )2i
g : {0, 1}2 → {0, 1}
bitwise operator ϕg
P
ϕg (x, y) = bi=0 g(ai , bi)2i
:. 4
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g
ϕ
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2.3.5 Goals
bool
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2.3.6 Exercises
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oo
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x
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rq
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int main ()
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int a ;
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bool b = a * 3 > a + 4 && !( a >= 5);
std :: cout < < (! b || ++ a > 4) < < " .\ n";
return 0;
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unsigned int a;
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bool c = a < 1 || b != 0 && 2 * a / ( a - 1) > 2;
std :: cout < < c < < "\n" ;
}
return 0;
2.3.7 Challenges
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% & +
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(
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(0, 1))), 1),
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eval
./ eval |./ zero |./ zero |./ one |./ and |./ not |./ or |./ one |./ and
|
./ eval |./ zero |./ one |./ or
"
*
+
+ # &
!
!
p
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x
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rq