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28 Tests and Conditionals

28Tests and Conditionals
Is 2+2 equal to 4? Let’s ask the Wolfram Language.
Test whether 2+2 is equal to 4:
2 + 2 == 4
 
We can also test whether 2×2 is greater than 5. We do that using >.
Test whether 2×2 is greater than 5:
2*2 > 5
 
The function If lets you choose to give one result if a test is True, and another if it’s False.
Since the test gives True, the result of the If is x:
If[2 + 2 == 4, x, y]
 
By using a pure function with /@, we can apply an If to every element of a list.
If an element is less than 4, make it x, otherwise make it y:
If[# < 4, x, y] & /@ {1, 2, 3, 4, 5, 6, 7}
 
You can also test for less than or equal using , which is typed as <=.
If an element is less than or equal to 4, make it x; otherwise, make it y:
If[# <= 4, x, y] & /@ {1, 2, 3, 4, 5, 6, 7}
 
This makes an element x only if it is equal to 4:
If[# == 4, x, y] & /@ {1, 2, 3, 4, 5, 6, 7}
 
You can test whether two things are not equal using , which is typed as !=.
If an element is not equal to 4, make it x; otherwise, make it y:
If[# != 4, x, y] & /@ {1, 2, 3, 4, 5, 6, 7}
 
Select elements in the list that are greater than 3:
Select[{1, 2, 3, 4, 5, 6, 7}, # > 3 &]
 
Select elements that are between 2 and 5:
Select[{1, 2, 3, 4, 5, 6, 7}, 2 <= # <= 5 &]
 
Beyond size comparisons like <, > and ==, the Wolfram Language includes many other kinds of tests. Examples are EvenQ and OddQ, which test whether numbers are even or odd. (The “Q” indicates that the functions are asking a question.)
4 is an even number:
EvenQ[4]
 
Select even numbers from the list:
Select[{1, 2, 3, 4, 5, 6, 7, 8, 9}, EvenQ[#] &]
 
In this case, we don’t need the explicit pure function:
Select[{1, 2, 3, 4, 5, 6, 7, 8, 9}, EvenQ]
 
IntegerQ tests whether something is an integer; PrimeQ tests whether a number is prime.
Select prime numbers:
Select[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, PrimeQ]
 
Sometimes we need to combine tests. && represents “and”, || represents “or” and ! represents “not”.
Select elements of the list that are both even and greater than 2:
Select[{1, 2, 3, 4, 5, 6, 7}, EvenQ[#] && # > 2 &]
 
Select[{1, 2, 3, 4, 5, 6, 7}, EvenQ[#] || # > 4 &]
 
Select elements that are not either even or greater than 4:
Select[{1, 2, 3, 4, 5, 6, 7}, ! (EvenQ[#] || # > 4) &]
 
The space between letters isn’t a letter; nor is “!”:
{LetterQ["a"], LetterQ["bc"], LetterQ["a b"], LetterQ["!"]}
 
Turn a string into a list of characters, then test which are letters:
LetterQ /@ Characters["30 is the best!"]
 
Select the characters that are letters:
Select[Characters["30 is the best!"], LetterQ]
 
Select letters that appear after position 10 in the alphabet:
Select[Characters["30 is the best!"], LetterQ[#] && LetterNumber[#] > 10 &]
 
You can use Select to find words in English that are palindromes, meaning that they are the same if you reverse them.
Select[WordList[], StringReverse[#] == # &]
 
MemberQ tests whether something appears as an element, or member, of a list.
5 appears in the list {1, 3, 5, 7}:
MemberQ[{1, 3, 5, 7}, 5]
 
Select[Range[100], MemberQ[IntegerDigits[#], 2] &]
 
ImageInstanceQ is a machine-learning-based function that tests whether an image is an instance of a particular kind of thing, like a cat.
Test if an image is of a cat:
ImageInstanceQ[\!\(\* GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJxkvYd3m1ea5jlndvec3T1nz+7Mdvdsd9dO91Rwt7uquhy6ykG2y7ZsS5Yl WTlRzDlHMeccQQQigyAyiEAQIIgMkGDOlCgqR1t2uXLP7M4/sM97LwBrZs95 fc/Fx48UTf6+B8/73vde/qecynMF//bf/Jt/U/s/YziX3fBRTU120/n/Ay8u VtQWF1bk531eUZdfmF/zds7/gIt2/Pdf8R/Nl7WtcU3jqrZ5ZbJxSd2AWKFo XFLWLyhqo9KyhYmy+ER5RFQcFRfFJKWL0vK4rAIXl+TVy/K6uLRmWUHBry/J K3EPbg6NFwZH831Def7h3PmhTP9IdnC00DuQ4xnI9g7kefpyXN2Zc/3Zru50 e/tVZ/u16ZYLlqZz0y3nHB1XplsvW5ovGurOGG+ctTactzRemG6+ZG+9Ym46 Z++4aGs952i/YGs97+q64u655uq6hk+Z7cmY6bo+25U515vj6cuY7Ut3D2X5 hAUU4qLARElAWhSWl8c01YvaGyuGpvhU/eJkw5K+ec3YvmHtWrd0bk33rVn7 NmwDW/aR7ZmxPZdod1a065FsucU8tj2SvTn5rkd24FcfBLS3gtrbYcNhxHhv wXI7aroXn364ZHvARh73F63Jl1aMj5btiMcrDoz4EOLh4vS9qPF+zPRgwXxv wXR/0fwgbrkbMeEi4m7EcCc8dRjS3o3o7oUNd0P6O2H9YUiHSN2QGMOG+xHj naDuMDB14KM4DOgPfJM3fZr9eYzaPY96z6Pcn1Pd9Kpveicx35mV77oVuIL/ o5vzSn5la0a+7ZLx2JqRrtlEFNbxNYtoyTC2YBhdMgsRyxZhXD8aNwwvG0YW p4YXpobxclE3GtePxU3jq9OSjZmJrVkZ+7cobs1rEPj28L9w26+9E6J4uGC8 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Select images of cats:
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Entity["Species", "Species:FelisCatus"]] &]
 
Here’s a geographic example of Select: find which cities in a list are less than 3000 miles from San Francisco.
Select cities whose distance from San Francisco is less than 3000 miles:
Select[{Entity["City", {"London", "GreaterLondon", "UnitedKingdom"}], Entity["City", {"NewYork", "NewYork", "UnitedStates"}], Entity["City", {"Tokyo", "Tokyo", "Japan"}], Entity["City", {"Chicago", "Illinois", "UnitedStates"}]}, GeoDistance[#, Entity["City", {"SanFrancisco", "California", "UnitedStates"}]] < Quantity[3000, "Miles"] &]
 
28.1Test whether 123^321 is greater than 456^123. »
Expected output:
Out[]=
28.2Get a list of numbers up to 100 whose digits add up to less than 5. »
Expected output:
Out[]=
28.3Make a list of the first 20 integers, with prime numbers styled red. »
Expected output:
Out[]=
28.4Find words in WordList[ ] that both begin and end with the letter “p”. »
Sample expected output:
Out[]=
28.5Make a list of the first 100 primes, keeping only ones whose last digit is less than 3. »
Expected output:
Out[]=
28.6Find Roman numerals up to 100 that do not contain “I”. »
Expected output:
Out[]=
28.7Get a list of Roman numerals up to 1000 that are palindromes. »
Expected output:
Out[]=
28.8Find names of integers up to 100 that begin and end with the same letter. »
Expected output:
Out[]=
28.9Get a list of words longer than 15 characters from the Wikipedia article on words. »
Sample expected output:
Out[]=
28.10Starting from 1000, divide by 2 if the number is even, and compute 3#+1& if the number is odd; do this repeatedly 200 times (Collatz problem). »
Expected output:
Out[]=
28.11Make a word cloud of 5-letter words in the Wikipedia article on computers. »
Sample expected output:
Out[]=
28.12Find words in WordList[ ] whose first 3 letters are the same as their last 3 read backward, but where the whole string is not a palindrome. »
Sample expected output:
Out[]=
28.13Find all 10-letter words in WordList[ ] for which the total of LetterNumber values is 100. »
Sample expected output:
Out[]=
+28.1Make a table of integers up to 25 where every integer ending in 3 is replaced with 0. »
Expected output:
Out[]=
+28.2Use Table and If to make a 5×5 array that is 1 on its leading diagonal, and 0 otherwise.  »
Expected output:
Out[]=
+28.3Get a list of numbers up to 1000 that are equal to 1 mod both 7 and 8. »
Expected output:
Out[]=
+28.4Make a list of numbers up to 100, where multiples of 3 are replaced by Black, multiples of 5 by White and multiples of 3 and 5 by Red»
Expected output:
Out[]=
+28.5Use Select to get a list of planets whose mass is larger than Earth. »
Expected output:
Out[]=
+28.6Make a 50×50 array plot in which a square at position i, j is black if Mod[i, j]==0»
Expected output:
Out[]=
+28.7Make a 100×100 array plot in which a square is black if the values of both its x and y positions do not contain a 5. »
Expected output:
Out[]=
Because = means something else in the Wolfram Language. You’ll get very strange results if you use = instead of == by mistake. (= is for assigning values of variables.) To avoid possible confusion, == is often read as “double equals”.
Why is “and” written as &&, not &?
Because & means other things in the Wolfram Language. For example it’s what ends a pure function.
How are ==, >, &&, etc. interpreted?
== is Equal, (!=) is Unequal, > is Greater, is GreaterEqual, < is Less, && is And, || is Or and ! is Not.
When do I need to use parentheses with &&, ||, etc.?
There’s an order of operations that’s a direct analog of arithmetic. && is like ×, || is like +, and ! is like . So !p&&q means “(not p) and q”; !(p&&q) means “not (p and q)”.
What’s special about “Q” functions?
They ask a question that normally has an answer of True or False.
What are some other “Q” functions?
Is there a better way to find real-world entities with certain properties than using Select?
Yes. You can do things like Entity["Country", "Population"GreaterThan[]] to find “implicit entities”, then use EntityList to get explicit lists of entities.
  • True and False are typically called Booleans in computer science, after George Boole from the mid-1800s. Expressions with &&, ||, etc. are often called Boolean expressions.
  • In the Wolfram Language, True and False are symbols, and are not represented by 1 and 0 as in many other computer languages.
  • If is often called a conditional. In If[test, then, else], the then and else aren’t computed unless the test says their condition is met.
  • PalindromeQ directly tests if a string is a palindrome.
  • In the Wolfram Language, x is a symbol (see Section 33) that could represent anything, so x==1 is just an equation, that isn’t immediately True or False. x===1 (“triple equals”) tests whether x is the same as 1, and since it isn’t, it gives False.
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