Booleans and Conditionals

In this lab, you will have the opportunity to explore a number of issues relating to predicates, Boolean values, and conditional operations.

Procedures covered in this lab include:

• Type predicates: boolean?, integer?, list?, null?, number?, pair?, procedure?, symbol?
• Equality comparators: =, eq?, eqv?, equal?,
• Numeric comparators: < (strictly less than), <= (less than or equal to), = (equal to), >= (greater than or equal to), > (strictly greater than)
• Boolean operations: and, or, not

• Exercises
  • Exercise 0: Preparation
  • Exercise 1: Type Predicates
  • Exercise 2: Empty lists
  • Exercise 3: Equality
  • Exercise 4: Reflection
  • Exercise 5: What is not?
  • Exercise 6: What is not? Revisited
  • Exercise 7: Combining Boolean Values
  • Exercise 8: Ranges
  • Exercise 9: Exploring and and or
  • Exercise 10: What is it?
  • Exercise 11: Between, Revisited
  • Exericse 12: Triangulation
  • Exercise 13: Being Neighborly
  • Exercise 14: Non-Boolean Tests
  • Exercise 15: Categorizing Lists
  • Exercise 16: Who Won?
  • Exercise 17: The Sphinx's Riddle
  • Exercise 18: Categorizing Data
• Notes
  • Notes on Problem 9

Exercises

Exercise 0: Preparation

You may find it helpful to rescan the readings on Boolean values and conditionals.

You may also want to rescan the reading on numbers.

After making sure that you're prepared, start DrScheme.

Exercise 1: Type Predicates

Fill in the following table (or as much as you think is appropriate). You need not fill int the whole table; simply do as much as you think gives you a good sense of the various predicates.

Exercise 2: Empty lists

Which of the following does Scheme consider an empty list?

• null
• 'null
• ()
• (list a)
• (list)
• 'nothing

Exercise 3: Equality

Consider the following definitions

(define alpha (list 'a 'b 'c))
(define beta (list 'a 'b 'c))
(define gamma alpha)

Determine which of the lists are eq?, eqv?, or equal?.

Exercise 4: Reflection

What, if anything, did you find surprising in the results of the previous exercises?

By looking at the Scheme report, see if you can find a pair of values that are equal in the sense of = but not in the sense of eqv.

Exercise 5: What is not?

a. What type is not?

b. What predicate would you use to verify your answer?

Exercise 6: What is not? Revisited

The symbol not is the name of something, as you determined in the preceding exercise. However, the symbol itself, considered as a datum, is not a procedure. Does Scheme agree with this classification? How could one ask Scheme whether the symbol not is a procedure?

Exercise 7: Combining Boolean Values

Fill in the following tables for each of the operations and and or.

and

or

Exercise 8: Ranges

a. Write a Boolean expression that determines if the value named by grade is between 0 and 100, inclusive.

b. Test that expression using different values of grade.

Exercise 9: Exploring and and or

a. Determine the value and returns when called with no parameters.

b. Explain why you think the designers of Scheme had and return that value.

c. Determine the value and returns when called with integers as parameters.

d. Explain why you think the designers of Scheme had and return that value.

e. Determine the value or returns when called with no parameters.

f. Explain why you think the designers of Scheme had or return that value.

g. Determine the value or returns when called with only integers as parameters.

h. Explain why you think the designers of Scheme had or return that value.

Exercise 10: What is it?

Define and test a Scheme predicate symbol-or-list? that takes one argument and returns #t if the argument is either a symbol or a list, #f if it is neither.

Exercise 11: Between, Revisited

Define and test a Scheme predicate between? that takes three arguments, all real numbers, and determines whether the second one lies strictly between the first and third (returning #t if it is, #f if it is not). For example, 6 lies strictly between 5 and 13, so both (between? 5 6 13) and (between? 13 6 5) should have the value #t.

Exericse 12: Triangulation

Three line segments can be assembled into a triangle if, and only if, the length of each of them is less than the sum of the lengths of the other two. Define a Scheme predicate triangle? that takes three arguments, all positive real numbers, and determines whether line segments of those three lengths (assumed to be measured in the same units) could be assembled into a triangle.

You should try to define triangle? without if (although it is certainly acceptable to use if in the definition).

Exercise 13: Being Neighborly

Define and test a Scheme procedure neighbor that takes one argument, an integer, and returns the next higher integer if its argument is even, the next lower integer if its argument is odd. (Start by writing a comment that describes the purpose of the procedure.)

Exercise 14: Non-Boolean Tests

For each of the following expressions, guess what the output should be and then test it in Scheme.

a. (if #t 'aardvark 'zebra)

b. (if #f 'aardvark 'zebra)

c. (if (null? null) 'aardvark 'zebra)

d. (if (null? 'null) 'aardvark 'zebra)

e. (if (null? (list 'a 'b 'c)) 'aardvark 'zebra)

f. (if () 'aardvark 'zebra)

g. (if (list 'a 'b 'c) 'aardvark 'zebra)

h. (if 2 'aardvark 'zebra)

i. (if 'true 'aardvark 'zebra)

j. (if 'false 'aardvark 'zebra)

Exercise 15: Categorizing Lists

Define and test a Scheme procedure list-type that takes one argument, a list, and returns the symbol empty if the argument is the empty list and the symbol non-empty otherwise.

Exercise 16: Who Won?

Define and test a Scheme procedure report-victory that takes one argument, a real number, and returns the symbol won if that number is positive, the symbol lost if it is negative, and the string tied if it is zero.

a. Use if for all the tests within your procedure.

b. Use cond for all the tests within your procedure.

Exercise 17: The Sphinx's Riddle

As you may know, one of the famous riddles of the Sphinx goes something like the following:

What is it that walks upon four legs, then two legs, then three legs?

The answer is humans.

Write a Scheme predicate, legs, that, given someone's age, tells how many legs they walk upon. (You get to choose reasonable ages for the three phases of life.)

Exercise 18: Categorizing Data

Write and test a Scheme procedure, type, that returns the a symbol that represents the type of its argument. For example,

> (type 3)
integer
> (type '(1 2 3))
list
> (type "hello")
string
> (type 2/3)
rational
> (type 'type)
symbol
> (type type)
procedure
> (type 1.2)
real
> (type 3+4i)
complex

You need only cover the types in the example above, but you should cover all of those types.

Warning! As you've noted in the past, you should not be able to distinguish between real numbers and rational numbers. I would recommend that you use rational for exact reals/rationals and real for inexact reals/rationals.

Notes

Notes on Problem 9

(and) has value true because and has a value of true if none of the parameters have value false. Since it has no parameters, none are false.

(or) has value false because or has value false if none of the parameters is non-false. Since it has no parameters, none are non-false.