Computer Science Fundamentals (CS153 2004S)
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A Boolean value is a datum that reflects the outcome of a
single yesorno test. For instance, if one were to ask Scheme to
compute whether the empty list has five elements, it would be able
to determine that it does not, and it would signal this result
by displaying the Boolean value for no
or false
,
which is #f
. There is only one other Boolean value,
the one meaning yes
or true
, which is #t
.
These are called Boolean values
in honor of the logician George
Boole, who was the first to develop a satisfactory formal theory
of them. (Some folks now talk about fuzzy logic
that includes
values other than true
and false
, but that's beyond the
scope of this course.)
A predicate is a procedure that always returns a Boolean
value. A procedure call in which the procedure is a predicate
performs some yesorno test on its arguments. For instance, the
predicate number?

the question mark is part of the name of the procedure  takes one
argument and returns #t
if that argument is a number,
#f
if it does not. Similarly, the predicate even?
takes one argument, which
must be an integer, and returns #t
if the integer is even
and #f
if it is odd. The names of most Scheme predicates
end with question marks, and Grinnell's computer scientists recommend
this useful convention, even though it is not required by the rules
of the programming language. (If you ever notice that I've failed to
include a question mark in a predicate and you're the first to tell me,
I'll give you some extra credit.)
Scheme provides a few predicates that let you test the type
of value you're working with.
number?
tests
whether its argument is a number.
symbol?
tests whether
its argument is a symbol.
string?
tests whether
its argument is a string (you'll learn about strings in a few days).
procedure?
tests
whether its argument is a procedure.
boolean?
tests
whether its argument is a Boolean value.
list?
tests whether
its argument is a list. (Warning! It can be quite expensive
to determine whether or not somethign is a list.)
Scheme provides one basic predicate for working with lists (other
than the list?
predicate).
null?
tests whether its
argument is the empty listvalue.
Scheme provides a variety of predicates for testing equality.
eq?
tests whether
its two arguments are identical, in the very narrow sense of occupying
the same storage location in the computer's memory. In practice, this
is useful information only if at least one argument is known to be a
symbol, a Boolean value, or an integer.
eqv?
tests whether
its two arguments should normally be regarded as the same object(as the language standard declares). Note, however, that two lists can have the same elements without being
regarded as the same object. Also note that in Scheme's view the number 5, which is
exact, is not necessarily the same object as the number 5.0, which might be an approximation.
equal?
tests whether
its two arguments are the same or,
in the case of lists, whether they have the same contents.
Scheme also provides many numeric predicates.
=
tests whether its arguments,
which must all be
numbers, are numerically equal; 5 and 5.0 are numerically equal for this
purpose.
<
tests whether its arguments, which must all be
numbers, are in strictly ascending numerical order. (The
<
operation is one of the few builtin predicates
that does not have an accompanying question mark.)
>
tests whether its arguments,
which must all be numbers, are in strictly descending numerical order.
<=
tests whether its arguments, which must all be
numbers, are in ascending numerical order, allowing equality.
>=
tests whether its arguments, which must all be
numbers, are in descending numerical order, allowing equality.
even?
tests whether its
argument, which must be an integer, is even.
odd?
tests whether its argument,
which must be an integer, is odd.
zero?
tests whether its argument, which must be a number, is equal to zero.
positive?
tests whether
its argument, which must be a real number, is positive.
negative?
tests whether
its argument, which must be a real number, is negative.
Another useful Boolean procedure is not
, which takes one argument and returns
#t
if the argument is #f
and #f
if the argument is anything else. For example, one can test whether the
square root of 100 is unequal to the absolute value of negative twelve
by giving the command
(not (= (sqrt 100) (abs 12)))
If Scheme says that the value of this expression is #t
, then
the two numbers are indeed unequal.
Two other useful Boolean procedures are and
and or
.
Can you guess what they do?
The and
and
or
procedures
have simple
logical meanings (in particular, the and of a collection of Boolean
values is true if all are true and false if any value is false, the or
of a collection of Boolean values is true if any of the values is
true and false if all the values are false. For example,
> (and #t #t #t) #t > (and (< 1 2) (< 2 3)) #t > (and (odd? 1) (odd? 3) (odd? 5) (odd? 6)) #f > (and) #t > (or (odd? 1) (odd? 3) (odd? 5) (odd? 6)) #t > (or (even? 1) (even? 3) (even? 4) (even? 5)) #t > (or) #f
But and
and
or
can be used for
so much more. In fact, they can be used as control structures.
In an and
expression,
the expressions that follow the keyword and
are evaluated in succession
until one is found to have the value #f
(in which case the
rest of the expressions are skipped and the #f
becomes the
value of the entire and
expression) or all of the
expressions have been evaluated (in which case the value of the last
expression becomes the value of the and
expression). This
gives the programmer a way to combine several tests into one that will
succeed only if all of its parts succeed.
In an or
expression, the
expressions that follow the keyword or
are evaluated in succession
until one is found to have a value other than#f
(in which
case the rest of the expressions are skipped and this value becomes the
value of the entire or
expression) or all of the
expressions have been evaluated (the value of the
or
expression is #f
). This gives the
programmer a way to combine several tests into one that will succeed if
any of its parts succeeds.
Although most computer scientists, philosophers, and mathematicians prefer
the purity of dividing the world into true
and false
, Scheme
supports a somewhat more general separation. In Scheme, anything that is
not false is considered true. Hence, you can use expressions that return
values other than truth values wherever a truth value is expected. For
example,
> (and #t 1) 1 > (or 3 #t #t) 3 > (not 1) #f > (not (not 1)) #t
In these cases, and
returns the last parameter it encounters
(or false, if it encounters a false value) while or
returns
the first nonfalse parameter it encounters.
In the early stages of your Scheme programming, you should probably avoid such nonlogical uses of the logical operations.
Monday, 4 September 2000 [Sam Rebelsky]
Wednesday, 31 January 2001 [Sam Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2000F/Readings/andor.html
,
although that section was extended and updated.
Tuesday, 10 September 2002 [Samuel A. Rebelsky]
http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2002F/Readings/boolean.html
.
Tuesday, 27 January 2003 [Samuel A. Rebelsky]
(not (not 1))
(thanks ON).
http://www.cs.grinnell.edu/~rebelsky/Courses/CS153/2003S/Readings/boolean.html
.
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Disclaimer:
I usually create these pages on the fly
, which means that I rarely
proofread them and they may contain bad grammar and incorrect details.
It also means that I tend to update them regularly (see the history for
more details). Feel free to contact me with any suggestions for changes.
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.
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