Programming Across Disciplines

Home » Chapter 3 : Decisions
Booleans & Conditions

Overview
Logical Operators
Truth Tables
Booleans are Numeric
Comparison Operators

Overview

Boolean is a binary state that can only be one of two values, True or False. Boolean variables in Python can only hold one of those two values and conditions can only result in one of those two values. We use boolean in many scenarios in programming, such as repeating an action until some condition becomes True. We often associate boolean result values of True and False to Yes or No questions.

Here are some example questions and statements that we could implement in Python that would rely on boolean values:

Is today Monday?
Is the user employed?
Is this student's GPA greater than or equal to 3.5?
Repeat adding numbers until we run out of numbers.

On the surface this sounds easy, it's just Yes (True) or No (False). However, there are many nuances about "truth" in programming. So, we need to learn what True & False mean in Python, how we represent those values, how we evaluate those values, and how we handle those values in our code. We will get started here, and like many things in programming, we will keep adding to this idea as we proceed.

First let's remind ourselves about the Logical Operators as we'll need these in our discussion.

Operator	Description
and	The and operator compares the condition to its left with the condition to its right, if both conditions are True then the compound condition is True.
or	The or operator compares the condition to its left with the condition to its right, if either of the conditions is True then the compound condition is True.
not	The not operator reverses the result of the compound condition, so if the compound returns True the not operator will reverse it to False, and visa versa.

On the previous few pages, we explored decision statements: if, if-else, and if-elif-else. Decision statements are also called boolean expressions because their result result in either a True or False value. In those expressions, we use one or more of the operators listed above to create the question to which we want the True or False answer.

Let's use one of the questions above as an example: Is the user employed?

If you recall from our earlier discussion about Data and Data Types, I used an example of prompting a user for information about them. One of the questions was whether or not they are employed. When our program collects the input from the user, the variable that we use to store their answer to that question should be of type bool. In this way, we have their answer in a form that we can easily evaluate because the variable employed contains our bool value which contains either True or False. So, then we can write code like this:

if user == employed:
    # do something

This code example demonstrates the use of the bool variable in a decision statement condition. We can use it in this way because the variable employed will contain either only True or False.

Concept

Evaluation: I have used the term evaluate several times so far, but what does that mean in the context of programming? Evaluation in programming means that we examine the result of something and, usually, take some action based on the result of the evaluation. So, for example, if I write

x = 10
y = 2
z = x * y

How might I examine the value of z? I could print it, but what if I needed to know whether the value of z is above or below a certain value? I can evaluate that question as a decision (a.k.a. a conditional statement).

if z > 15:
# do something

With this if statement I am evaluating the value of z as a condition, in this example the condition of z > 15 will evaluate to True, so my # do something code will run.

Understanding the evaluation of boolean logic leads us to the concept of Truth Tables

The simplest truth table is a single variable that displays all possible boolean values for the variable, which are only True and False.

x
True
False

Since our first truth table contains only one variable, the only operation we can perform on it using the logical operators we know is not. The not operator reverses the truth value of a variable or expression. The second column now displays the result of using not on the variable.

x	not x
True	False
False	True

When we add a second variable to our truth table, we add the variable to its column in the table and enter its possible values given the problem scenario.

x	y
True	True
True	False
False	True
False	False

Now that we have two variables, we can use the other logical operators and and or. With our propositions and value columns set up, we add columns for each comparison we may need in the problem solution. In this example, we want to evaluate x and y and x or y. Then for each row, we apply the logical operators to the propositions in the first two columns and populate the appropriate cells in the comparison columns. For example, on the first row x = True and y = True, so we evaluate the comparision of x and y, that is, True and True, the result of which is True. So we enter True in the first row under x and y. We then continue with the remaining cells to fill out the rest of the table.

x	y	x and y
True	True	True
True	False
False	True
False	False

Once the table is fully populated we can use this table to prepare a code solution that will cover all possible results from the compound condition problem. So, we can transfer these logical results to a programming construct so that, for example, all four possible outcomes of x and y have a coded action as needed.

x	y	x and y	x or y
True	True	True	True
True	False	False	True
False	True	False	True
False	False	False	True

We are not limited to using just two variables in a truth table. Our problem scenario may require more than two boolean variables to evaluate. In these cases, truth tables are very useful to help us visualize the possible outcomes.

Based on this truth table, answer the following question:

w	x	y	z	w and x and y and z
True	False	True	True	False
True	True	True	True	True
True	True	False	True	False
False	True	False	True	False

Question 1

Explain the results displayed in the fifth column for this truth table where w and x and y and z.

Also, we are not limited to using just one logical operator at a time. Our problem scenario may require compound conditions as well. Compound conditions are several conditions that, when combined, return a single True/False result. In these cases, truth tables are very useful to help us visualize the possible outcomes. For example, consider this truth table:

Based on this truth table, answer the following question:

w	x	y	z	w and (x or y) and (z or w)
True	False	True	True	True
True	True	True	True	True
True	True	False	True	True
False	True	False	True	False

Question 2

Explain the results displayed in the fifth column for this truth table where w and (x or y) and (z or w).

First, like the previous problem above, there are four proposition variables, each of which hold a boolean value. In this case, we notice that our formula contains parenthesis surrounding two sets of variables. Due to order of operations, we always evaluate what is inside the parentisis first. So, taking the first row of the truth table as an example, if we substitute the True and False values in the formula we have this:

True and (False or True) and (True or True)

Evaluating the parenthsis first, we then have:

True and True and True

We can see now that our compound condition result is True.

And how about the last row? Let's step through it. We start with:

False and (True or False) and (True and False)

We evaluate the parenthsis first, which gives us:

False and True and True

Since the first proposition is False and our two remaining logical operators are and, the result of the compound condition is False. In order for this to be True, given the two and operators, all three propositions would have to be True.

We can see this in Python code like this:

w = True
  x = False
  y = True
  z = True
  print(w and (x or y) and (z or w))
  w = True
  x = True
  y = True
  z = True
  print(w and (x or y) and (z or w))
  w = True
  x = True
  y = False
  z = True
  print(w and (x or y) and (z or w))
  w = False
  x = True
  y = False
  z = True
  print(w and (x or y) and (z or w))

Output:

True
  True
  True
  False

Operator	Comparison
==	Equal
!=	Not Equal
>	Greater Than
<	Less Than
>=	Greater Than or Equal To
<=	Less Than or Equal To

Overview

Menu

Logical Operators

Concept

Concept

Truth Tables

Question 1

Question 2

Booleans are Numeric

Comparison Operators