Functions that Test Conditions

Here is a simple function that tests some condition about a number:

<#70686#>;; <#60589#><#2077#>is-5?<#2077#> <#2078#>:<#2078#> <#2079#>number<#2079#> <#2080#><#2080#><#2081#>-;SPMgt;<#2081#><#2082#><#2082#> <#2083#>boolean<#2083#><#60589#><#70686#>
<#70687#>;; to determine whether <#60590#><#2084#>n<#2084#><#60590#> is equal to <#60591#><#2085#>5<#2085#><#60591#><#70687#> 
<#2086#>(d<#2086#><#2087#>efine<#2087#> <#2088#>(is-5?<#2088#> <#2089#>n)<#2089#> 
  <#2090#>(=<#2090#> <#2091#>n<#2091#> <#2092#>5))<#2092#>

The function produces <#60592#><#2096#>true<#2096#><#60592#> if, and only if, its input is equal to <#60593#><#2097#>5<#2097#><#60593#>. Its contract contains one novel element: the word <#60594#><#2098#>boolean<#2098#><#60594#>. Just like <#60595#><#2099#>number<#2099#><#60595#>, <#60596#><#2100#>boolean<#2100#><#60596#> represents a class of values that is built into Scheme. Unlike <#60597#><#2101#>number<#2101#><#60597#>, <#60598#><#2102#>boolean<#2102#><#60598#> consists of just two values: <#60599#><#2103#>true<#2103#><#60599#> and <#60600#><#2104#>false<#2104#><#60600#>. Here is a slightly more interesting function with a Boolean output:

<#70688#>;; <#60601#><#2109#>is-between-5-6?<#2109#> <#2110#>:<#2110#> <#2111#>number<#2111#> <#2112#><#2112#><#2113#>-;SPMgt;<#2113#><#2114#><#2114#> <#2115#>boolean<#2115#><#60601#><#70688#>
<#70689#>;; to determine whether <#60602#><#2116#>n<#2116#><#60602#> is between 5 and 6 (exclusive)<#70689#> 
<#2117#>(d<#2117#><#2118#>efine<#2118#> <#2119#>(is-between-5-6?<#2119#> <#2120#>n)<#2120#> 
  <#2121#>(and<#2121#> <#2122#>(;SPMlt;<#2122#> <#2123#>5<#2123#> <#2124#>n)<#2124#> <#2125#>(;SPMlt;<#2125#> <#2126#>n<#2126#> <#2127#>6)))<#2127#>

It consumes a number and produces <#60603#><#2131#>true<#2131#><#60603#> if the number is between, but does not include, <#60604#><#2132#>5<#2132#><#60604#> and <#60605#><#2133#>6<#2133#><#60605#>. One good way to understand the function is to say that it describes the following interval on the number line:

<#72378#>#tex2html_wrap72654#<#72378#>

<#2147#>Boundaries<#2147#>:\ An interval boundary marked with ``('' or ``)'' is excluded from the interval; an interval boundary marked with ``['' or ``]'' is included.~<#60614#><#60614#>

The following third function from numbers to Boolean values represents the most complicated form of interval:

<#70692#>;; <#60615#><#2154#>is-between-5-6-or-over-10?<#2154#> <#2155#>:<#2155#> <#2156#>number<#2156#> <#2157#><#2157#><#2158#>-;SPMgt;<#2158#><#2159#><#2159#> <#2160#>boolean<#2160#><#60615#><#70692#>
<#70693#>;; to determine whether <#60616#><#2161#>n<#2161#><#60616#> is between 5 and 6 (exclusive) <#70693#> 
<#70694#>;; or larger than or equal to <#60617#><#2162#>10<#2162#><#60617#><#70694#> 
<#2163#>(d<#2163#><#2164#>efine<#2164#> <#2165#>(is-between-5-6-or-over-10?<#2165#> <#2166#>n)<#2166#> 
  <#2167#>(or<#2167#> <#2168#>(is-between-5-6?<#2168#> <#2169#>n)<#2169#> <#2170#>(;SPMgt;=<#2170#> <#2171#>n<#2171#> <#2172#>10)))<#2172#>

The function returns <#60618#><#2176#>true<#2176#><#60618#> for two portions of the number line:

<#72379#>#tex2html_wrap72656#<#72379#> The left part of the interval is the portion between, but not including, <#60629#><#2193#>5<#2193#><#60629#> and <#60630#><#2194#>6<#2194#><#60630#>; the right one is the infinite line starting at, and including, <#60631#><#2195#>10<#2195#><#60631#>. Any point on those two portions of the line satisfies the condition expressed in the function <#60632#><#2196#>is-between-5-6-or-over-10?<#2196#><#60632#>. All three functions test numeric conditions. To design or to comprehend such functions, we must understand intervals and combinations of intervals. The following exercises practice this important skill.
<#2199#>Exercise 4.2.1<#2199#> Translate the following five intervals on the real line into Scheme functions that accept a number and return <#60633#><#2202#>true<#2202#><#60633#> if the number is in the interval and <#60634#><#2203#>false<#2203#><#60634#> if it is outside:

the interval (3,7]:
<#72380#>#tex2html_wrap72658#<#72380#>
the interval [3,7]:
<#72381#>#tex2html_wrap72660#<#72381#>
the interval [3,9):
<#72382#>#tex2html_wrap72662#<#72382#>
the combination of (1,3) and (9,11):
<#72383#>#tex2html_wrap72664#<#72383#>
and the range of numbers <#2261#>outside<#2261#> of [1,3].
<#72384#>#tex2html_wrap72666#<#72384#>

Solution<#60679#><#60679#> <#2283#>Exercise 4.2.2<#2283#> Translate the following four Scheme functions into intervals on the line of reals:

<#2290#>1.<#2290#> <#2291#>(defin<#2291#><#2292#>e<#2292#> <#2293#>(in-interval-1?<#2293#> <#2294#>x)<#2294#>
         <#2295#>(and<#2295#> <#2296#>(;SPMlt;<#2296#> <#2297#>-3<#2297#> <#2298#>x)<#2298#> <#2299#>(;SPMlt;<#2299#> <#2300#>x<#2300#> <#2301#>0)))<#2301#> 
<#2302#>2.<#2302#> <#2303#>(defin<#2303#><#2304#>e<#2304#> <#2305#>(in-interval-2?<#2305#> <#2306#>x)<#2306#> 
         <#2307#>(or<#2307#> <#2308#>(;SPMlt;<#2308#> <#2309#>x<#2309#> <#2310#>1)<#2310#> <#2311#>(;SPMgt;<#2311#> <#2312#>x<#2312#> <#2313#>2)))<#2313#> 
<#2314#>3.<#2314#> <#2315#>(defin<#2315#><#2316#>e<#2316#> <#2317#>(in-interval-3?<#2317#> <#2318#>x)<#2318#> 
         <#2319#>(not<#2319#> <#2320#>(and<#2320#> <#2321#>(;SPMlt;=<#2321#> <#2322#>1<#2322#> <#2323#>x)<#2323#> <#2324#>(;SPMlt;=<#2324#> <#2325#>x<#2325#> <#2326#>5))))<#2326#>

Also formulate contracts and purpose statements for the four functions. Evaluate the following expressions by hand:

<#60680#><#2331#>(in-interval-1?<#2331#>\ <#2332#>-2)<#2332#><#60680#>
<#60681#><#2333#>(in-interval-2?<#2333#>\ <#2334#>-2)<#2334#><#60681#>
<#60682#><#2335#>(in-interval-3?<#2335#>\ <#2336#>-2)<#2336#><#60682#>

Show the important steps. Use the pictures to check your results.~

Solution<#60683#><#60683#> <#2343#>Exercise 4.2.3<#2343#> Mathematical equations in one variable are claims about an unknown number. For example, the quadratic equation

#displaymath72634#

is a claim concerning some unknown number x. For x = -1 the claim holds:

#displaymath72640#

For x = 1, it doesn't because

#displaymath72644#

and 4 is <#2345#>not<#2345#> equal to 0. A number for which the claim holds is called a <#2346#>solution<#2346#> to the equation. We can use Scheme to formulate equational conditions as a function. If someone then claims to have a solution, we can use the function to test whether the proposed solution is, in fact, a solution. Our running example corresponds to the function

<#70710#>;; <#60684#><#2351#>equation1<#2351#> <#2352#>:<#2352#> <#2353#>number<#2353#> <#2354#><#2354#><#2355#>-;SPMgt;<#2355#><#2356#><#2356#> <#2357#>boolean<#2357#><#60684#><#70710#>
<#70711#>;; to determine whether <#60685#><#2358#>x<#2358#><#60685#> is a solution for #tex2html_wrap_inline72646#<#70711#> 
<#2359#>(d<#2359#><#2360#>efine<#2360#> <#2361#>(equation1<#2361#> <#2362#>x)<#2362#> 
  <#2363#>(=<#2363#> <#2364#>(+<#2364#> <#2365#>(*<#2365#> <#2366#>x<#2366#> <#2367#>x)<#2367#> <#2368#>(+<#2368#> <#2369#>(*<#2369#> <#2370#>2<#2370#> <#2371#>x)<#2371#> <#2372#>1))<#2372#> <#2373#>0))<#2373#>

When we apply <#60686#><#2377#>equation1<#2377#><#60686#> to some number, we get <#60687#><#2378#>true<#2378#><#60687#> or <#60688#><#2379#>false<#2379#><#60688#>:

  <#2384#>(equation1<#2384#> <#2385#>-1)<#2385#>
<#2386#>=<#2386#> <#2387#>true<#2387#>

and

  <#2395#>(equation1<#2395#> <#2396#>+1)<#2396#>
<#2397#>=<#2397#> <#2398#>false<#2398#>

Translate the following equations into Scheme functions:

#tex2html_wrap_inline72648#
#tex2html_wrap_inline72650#
#tex2html_wrap_inline72652#

Determine whether <#60689#><#2404#>10<#2404#><#60689#>, <#60690#><#2405#>12<#2405#><#60690#>, or <#60691#><#2406#>14<#2406#><#60691#> are solutions of these equations.

Solution<#60692#><#60692#> <#2412#>Exercise 4.2.4<#2412#> Equations are not only ubiquitous in mathematics, they are also heavily used in programming. We have used equations to state what a function should do with examples, we have used them to evaluate expressions by hand, and we have added them as test cases to the <#2414#>Definitions<#2414#> window. For example, if our goal is to define <#72165#><#70712#><#2415#>Fahrenheit<#2415#><#60693#><#2416#><#2416#><#2417#>-;SPMgt;<#2417#><#2418#><#2418#><#60693#><#2419#>Celsius<#2419#><#70712#><#72165#>, we might add have added our examples as test cases as follows:

<#2424#>(<#2424#><#70713#><#2425#>Fahrenheit<#2425#><#60694#><#2426#><#2426#><#2427#>-;SPMgt;<#2427#><#2428#><#2428#><#60694#><#2429#>Celsius<#2429#><#70713#> <#2430#>32)<#2430#>
<#2431#>=<#2431#> 
<#2432#>0<#2432#>

and

<#2440#>(<#2440#><#70714#><#2441#>Fahrenheit<#2441#><#60695#><#2442#><#2442#><#2443#>-;SPMgt;<#2443#><#2444#><#2444#><#60695#><#2445#>Celsius<#2445#><#70714#> <#2446#>212)<#2446#>
<#2447#>=<#2447#> 
<#2448#>100<#2448#>

After clicking the <#2452#>Execute<#2452#> button we can compare the numbers to the left and right of <#60696#><#2453#>=<#2453#><#60696#>. If they are equal, we know our function works. As our results become more and more complex, comparing values becomes more and more tedious. Using <#60697#><#2454#>=<#2454#><#60697#>, we can instead translate these equations into claims:

<#2459#>(=<#2459#> <#2460#>(<#2460#><#70715#><#2461#>Fahrenheit<#2461#><#60698#><#2462#><#2462#><#2463#>-;SPMgt;<#2463#><#2464#><#2464#><#60698#><#2465#>Celsius<#2465#><#70715#> <#2466#>32)<#2466#>
   <#2467#>0)<#2467#>

and

<#2475#>(=<#2475#> <#2476#>(<#2476#><#70716#><#2477#>Fahrenheit<#2477#><#60699#><#2478#><#2478#><#2479#>-;SPMgt;<#2479#><#2480#><#2480#><#60699#><#2481#>Celsius<#2481#><#70716#> <#2482#>212)<#2482#>
   <#2483#>100)<#2483#>

Now, if all claims evaluate to <#60700#><#2487#>true<#2487#><#60700#>, we know that our function works for the specified examples. If we see a <#60701#><#2488#>false<#2488#><#60701#> anywhere, something is still wrong. Reformulate the test cases for exercises~#exf2c#2489>, #exusdm#2490>, #extriangle#2491>, and~#exconvert3#2492> as claims.

<#2493#>Testing<#2493#>:\ Writing tests as claims is good practice, though we need to know more about equality to develop good automatic tests. To do so, we resume the discussion of equality and testing in section~#secequaltest#2494>.~ Solution<#60702#><#60702#>