A Boolean value is a datum that reflects the outcome of a
single yes-or-no 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
#f. There is only one other Boolean value,
the one meaning
true, which is
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
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 yes-or-no test on its arguments. For instance, the
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
#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
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
null?tests whether its argument is the
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 built-in 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
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
the two numbers are indeed unequal.
Two other useful Boolean procedures are
Can you guess what they do?
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
or can be used for
so much more. In fact, they can be used as control structures.
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
gives the programmer a way to combine several tests into one that will
succeed only if all of its parts succeed.
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
#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
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
> (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
the first non-false parameter it encounters.
In the early stages of your Scheme programming, you should probably avoid such non-logical 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]
Tuesday, 27 January 2003 [Samuel A. Rebelsky]
(not (not 1))(thanks ON).
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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