1. Boolean Values

Python has two values of the bool type, written True and False. These are called logical values
or Boolean values, named after 19th century mathematician George Boole.

In [1]:
True          # show the value
Out[1]:
True
In [2]:
type(True)    # check the type name
Out[2]:
bool
In [3]:
False         # show the value
Out[3]:
False
In [4]:
type(False)   # check the type name
Out[4]:
bool

Careful, the two values are written as uppercase, and you'll get an error if they are misspelled:

In [5]:
true
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-5-724ba28f4a9a> in <module>
----> 1 true

NameError: name 'true' is not defined
In [6]:
false
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-6-b73d74fcede9> in <module>
----> 1 false

NameError: name 'false' is not defined

2. Relational Operators

We have seen arithmetic operators that produce as output numerical values. Today, we'll see relational operators that
produce as output Boolean values. Relational operators are used to compare two values.

Below try to first guess the values before running the cell.

In [7]:
3 < 5
Out[7]:
True
In [8]:
3 < 2
Out[8]:
False
In [9]:
3 > 2
Out[9]:
True
In [10]:
5 == 5
Out[10]:
True
In [11]:
5 >= 5
Out[11]:
True
In [12]:
6 <= 5
Out[12]:
False

Note: == is pronounced "equals" and != is pronounced "not equals". This is why we distinguish the pronunciation
of the single equal sign = as "gets", which is assignment and nothing to do with mathematical equality!

Relational operators can also be used to compare strings (in dictionary order).

In [13]:
'bat' < 'cat'
Out[13]:
True
In [14]:
'bat' < 'ant'
Out[14]:
False
In [15]:
'bat' == 'bat'
Out[15]:
True
In [16]:
'bat' < 'bath'
Out[16]:
True
In [17]:
'Cat' < 'bat'
Out[17]:
True

EXPLANATION: How does this comparison of string values work? Python starts by comparing the first character of each string to one another. For example "b" with "c". Because the computer doesn't know anything about letters, it converts everything into numbers. Each character has a numerical code that is summarized in this table of ASCII codes. In Python, we can look up the ASCII code via the Python built-in function ord:

In [18]:
print(ord('a'), ord('b'), ord('c'))
97 98 99

As you can see, the value for 'b', 98, is smaller than the value for 'c', 99, thus, 'b' < 'c'. Once two unequal characters are found, Python stops comparing the other characters, because there is no point in continuing. However, if characters are the same, like in 'bat' and 'bath', the comparisons continue until the point in which something that differs is found. In this case, there is an extra 't', making 'bath' greater in value than 'bat'.

Uppercase vs. Lowercase: Counterintuitively, it turns out, the upppercase letters are internally represented with smaller numbers than lowercase letters. See the ASCII table and the examples below:

In [19]:
print(ord('A'), ord('a'))
65 97
In [20]:
print(ord('B'), ord('b'))
66 98

This explains why the world 'Cat' is smaller than the word 'cat'.

3. Logical Operators

There are three logical operators: not, and, or, which are applied on expressions that are already evaluated as boolean values.

not

not expression evaluates to the opposite of the truth value of expression

In [21]:
not (3 > 5) # parentheses not necessary - relational operators have higher precedence
Out[21]:
True
In [22]:
not (3 == 3)
Out[22]:
False

and

exp1 and exp2 evaluates to True iff both exp1 and exp2 evaluate to True.

In [23]:
True and True
Out[23]:
True
In [24]:
True and False
Out[24]:
False
In [25]:
(3 < 5) and ('bat' < 'ant')
Out[25]:
False
In [26]:
(3 < 5) and ('bat' < 'cat')
Out[26]:
True

or

exp1 or exp2 evaluates to True iff at least one of exp1 and exp2 evaluate to True.

In [27]:
True or True
Out[27]:
True
In [28]:
True or False
Out[28]:
True
In [29]:
(3 > 5) or ('bat' < 'cat')
Out[29]:
True
In [30]:
(3 > 5) or ('bat' < 'ant')
Out[30]:
False

4. Combining logical operators

You can assign booleans to variables just like any other value. The variables below represent whether I like particular genres of music.

In [31]:
# change these as you like to experiment
pop = False
rap = True
hiphop = True
In [32]:
not pop
Out[32]:
True
In [33]:
pop and rap
Out[33]:
False
In [34]:
rap and hiphop
Out[34]:
True
In [35]:
pop or hiphop
Out[35]:
True
In [36]:
hiphop or rap
Out[36]:
True

What are the order of operations here? Do we compute the operator and first or or?

In [37]:
rap or hiphop and pop
Out[37]:
True

and takes precendence over or, so the above expression is the same as

In [38]:
rap or (hiphop and pop)
Out[38]:
True

but not the same as

In [39]:
(rap or hiphop) and pop
Out[39]:
False

5. Predicates

Definition: A predicate is simply any function that returns a boolean value.

In [40]:
def doILikeMissyElliott(rap, hiphop, pop):
    return rap or hiphop and pop
In [41]:
doILikeMissyElliott(False, True, True)
Out[41]:
True
In [42]:
doILikeMissyElliott(True, True, False)
Out[42]:
True
In [43]:
doILikeMissyElliott(False, False, True)
Out[43]:
False

Usually, the function body will contain a complex expression combining relational and logical expressions, as the following examples show:

In [44]:
def isFaculty(s):
    return (s == 'Eni' or s == 'Carolyn'
            or s == 'Andy' or s == 'Lyn'
            or s == 'Sohie' or s == 'Peter')
In [45]:
isFaculty('Carolyn')
Out[45]:
True
In [46]:
isFaculty("andy")
Out[46]:
False

Note: Explain the result of the last cell. Is that what you expected?

Expressing intervals of numbers: We can combine relational expressions to create intervals of numbers that fulfill certain criteria. Below is a predicate that checks if a value is within a given interval of numbers.

In [47]:
def isBetween(n, lo, hi):
    """determines if n is between lo and hi"""
    return (lo <= n) and (n <= hi)

More fun with Math: Is a number divisible by a factor? Is it even?

In [48]:
def isDivisibleBy(num, factor):
    """determines if num is divisible by factor"""
    return (num % factor) == 0 # notice the remainder operator 

def isEven(n):
    """determines if n is even"""
    return isDivisibleBy(n, 2)

print('Is 3774 divisible by 11?', isDivisibleBy(3774, 11))
print('Is 473 even?', isEven(473))
Is 3774 divisible by 11? False
Is 473 even? False

Is n a prime integer less than 100? In the solution below, notice how we split the long expression across mutliple lines for readability, and wrap all multiline expressions in parentheses.

In [49]:
# Version with continuation characters for multiline expressions
def isSmallPrime(n):
    return (isinstance(n, int)  # is n an integer?
            and (n > 1) and (n < 100)  # is n between 1 and 100?
            and (n==2 or n==3 or n==5 or n==7  # is n 2, 3, 5, or 7?
                 or not (isDivisibleBy(n,2)    #is n divisible by 2, 3, 5, or 7?
                         or isDivisibleBy(n,3)
                         or isDivisibleBy(n,5)
                         or isDivisibleBy(n,7))))
In [50]:
isSmallPrime(23)
Out[50]:
True
In [51]:
isSmallPrime(42)
Out[51]:
False

When multiline expressions are not wrapped in parentheses, you must use the special continuation character \ at the end of each line. No character other than a newline can come after the \.

In [52]:
# Version with continuation characters for some multiline expressions
def isSmallPrime(n):
    return isinstance(n, int) \
           and (n > 1) and (n < 100) \
           and (n==2 or n==3 or n==5 or n==7    # No need for \ because of parentheses
                or not (isDivisibleBy(n,2)
                        or isDivisibleBy(n,3)
                        or isDivisibleBy(n,5)
                        or isDivisibleBy(n,7)))

The version below is an alternative solutions that uses De Morgan's laws.

In [53]:
# alternate version using De Morgan's laws
def isSmallPrime2(n):
    return (isinstance(n, int)
            and (n > 1) and (n < 100)  # is n between 1 and 100?
            and (n==2 or n==3 or n==5 or n==7  # is n 2, 3, 5, or 7?
                 or (not isDivisibleBy(n,2)   #is n divisible by 2, 3, 5, or 7?
                     and not isDivisibleBy(n,3)
                     and not isDivisibleBy(n,5)
                     and not isDivisibleBy(n,7))))

6. Exercise: Define these predicates

Write the following four predicates.
Remember: a predicate is simply a function that returns a Boolean value.

isFreezing: is the temperature in Fahrenheit at or below freezing?
isLongName: is a name longer than 20 characters?
isVowel: is a character a vowel?
startsWithVowel: does a string start with a vowel?

Do not use conditional statements to solve these problems. Only relational or logical expressions!

In [54]:
# 1. isFreezing
# Your code here
def isFreezing(temp):
    return temp <= 32
In [55]:
isFreezing(10)
Out[55]:
True
In [56]:
isFreezing(75)
Out[56]:
False
In [57]:
# 2. isLongName
# Your code here
def isLongName(name):
    return len(name) > 20
In [58]:
isLongName('Wellesley')
Out[58]:
False
In [59]:
isLongName('Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch')
Out[59]:
True

(Above is the name of a village in Wales; you can hear it pronounced in this video.)

In [60]:
# 3. isVowel

# There are multiple possible solutions for this problem
# Your code here
def isVowel(letter):
    return letter.lower() in 'aeiou'
In [61]:
isVowel('e')
Out[61]:
True
In [62]:
isVowel('U')
Out[62]:
True
In [63]:
isVowel('b')
Out[63]:
False

Note: if s is a string, s[0] returns the first character of s.

In [64]:
# 4. startsWithVowel
# Your code here
def startsWithVowel(word):
    return isVowel(word[0])
In [65]:
startsWithVowel('Esmeralda')
Out[65]:
True
In [66]:
startsWithVowel('bravery')
Out[66]:
False

7. Drills: Booleans, Relational Operators, and Logical Operators

Write an expression that uses 1, 5, and < to produce the value True, and store the result in the variable x

In [67]:
# Your code here:
x = 1 < 5
In [68]:
assert x == True
print('Success!')
Success!

Write an expression that uses 1, 5, and < to produce the value False and store the result in the variable y

In [69]:
# Your code here:
y = 5 < 1
In [70]:
assert y == False
print('Success!')
Success!

Write an expression that uses True and not to produce the value False and store the result in the variable nope

In [71]:
# Your code here:
nope = not True
In [72]:
assert nope == False
print('Success!')
Success!

Write an expression that uses 1, 5, not, == to produce the value True and store the result in the variable z

In [73]:
# Your code here:
z = not 1 == 5
In [74]:
assert z == True
print('Success!')
Success!

Write an expression that uses two boolean values and the operator and to produce the value True and store the result in the variable sure

In [75]:
# Your code here:
sure = True and True
In [76]:
assert sure == True
print('Success!')
Success!

Write an expression that uses two boolean values and the operator or to produce the value True and store the result in the variable whyNot

In [77]:
# Your code here:
whyNot = True or False
In [78]:
assert whyNot == True
print('Success!')
Success!

Write an expression that uses two boolean values and the operator and to produce the value False and store the result in the variable becauseISaidSo

In [79]:
# Your code here:
becauseISaidSo = True and False
In [80]:
assert becauseISaidSo == False
print('Success!')
Success!

Write an expression that uses two boolean values and the operator or to produce the value False and store the result in the variable fairEnough

In [81]:
# Your code here:
fairEnough = False or False