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Unit 9: Logical Expression

Learning Objectives

After this unit, students should:

  • be able to read and write logical expressions in C using various logical operators, including ==, <, <=, <, >=, !=, &&, || and !;
  • be aware of the bool type and its values true and false, and the need to include stdbool.h to use it in C;
  • be aware that in CS1010, we must never use int to indicate a true/false value;
  • be aware of short-circuiting in logical expression;
  • be aware of the CS1010's convention of naming a boolean variable with the prefix is_, has_, or can_;
  • be able to write a logical expression in appropriate order to exploit short-circuiting toward more efficient code

Representing a Boolean Value

You have seen some basic logical expressions in Unit 8. n == 0, score >= 5, y > x, are all logical expressions. They evaluate to be either true or false.

We call a type that can contain either true or false a Boolean data type, named after George Boole, a mathematician.

The Boolean data type in C is named bool. It can hold two values: true or false. All three of bool, true, and false are keywords introduced in modern C. To use them, you need to include the file stdbool.h at the top of your program.

Using bool is considered a cleaner way of representing true and false in C. Classically, C defines the numeric value 0 to be false, and everything else to be true. So, you can write code like this:

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// x and y are long.
long is_diff = x - y;
if (is_diff) {
  cs1010_println_string("x and y store different values.");
}

The above is harder to understand and should be avoided. A cleaner way is to write:

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bool is_diff = x != y;
if (is_diff) {
  cs1010_println_string("x and y store different values.");
}

Although not required by C, we will name a bool variable with a prefix is_, has_, or can_, as a convention.

The code above can also be written as: 

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bool is_diff = x != y;
if (is_diff == true) {
  cs1010_println_string("x and y store different values.");
}

The comparison with true is redundant, however, and should be skipped.

Logical Operators

Just like we can perform arithmetic operations on integers and real numbers, we can perform logical operations on boolean values. These operations allow us to write complex logical expressions.

Consider the example problem: Write a function that, given the birth year of a person, determine if he or she belongs to Generation Z, defined as someone whose birth is between 1995 and 2005, inclusive.

We can write the function as follows using what we have known: 

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bool is_gen_z(long birth_year)
{
  if (birth_year >= 1995) {
    if (birth_year <= 2005) {
      return true;
    }
  }
  return false;
}

To be in Generation Z, both conditions birth_year >= 1995 and birth_year <= 2005 must be true. We can use the logical AND && operator to simplify the code above to:

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bool is_gen_z(long birth_year)
{
  if ((birth_year >= 1995) && (birth_year <= 2005)) {
    return true;
  }
  return false;
}

or simply: 

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bool is_gen_z(long birth_year)
{
  return ((birth_year >= 1995) && (birth_year <= 2005));
}

The AND operator, &&, evaluates to true if and only if both operands are true.

Common Error

A common mistake by a new C programmer is to write 1995 <= birth_year <= 2005 as the logical expression. Unfortunately, in C, we cannot chain the comparison operators together.

What if we want to write a function to determine if someone is NOT part of Generation Z? This means that they are born either before 1995 or after 2005. To have an expression that evaluates to true if at least one of two expressions is true, we can use the OR operator, ||.

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bool is_not_gen_z(long birth_year)
{
  return ((birth_year < 1995) || (birth_year > 2005));
}

Generally, we prefer to write functions that check for the positives, as it is generally easier to think in terms of the positives. So the example is_not_gen_z above is for illustration purposes only, we do not encourage you to write functions that check for the negatives. In any case, if we want to check if someone is not a Generation Z, we can use the ! NOT operator.

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if (!is_gen_z(birth_year)) {
     :
}

The ! operator can be used as part of the boolean expression:

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bool is_not_gen_z(long birth_year)
{
  return !((birth_year >= 1995) && (birth_year <= 2005));
}

The table below summarizes the logical operations &&, || and !:

a b a && b a || b !a
true true true true false
true false false true false
false true false true true
false false false false true

Short-Circuiting

When evaluating the logical expressions that involve && and ||, C uses "short-circuiting". If the program already knows, for sure, that a logical expression is true or false, there is no need to continue the evaluation. The corresponding true or false value will be returned.

Consider the following: 

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bool is_gen_z(long birth_year)
{
  return ((birth_year >= 1995) && (birth_year <= 2005));
}

If the argument birth_year is 1970, then, the expression (birth_year >= 1995) already evaluates to false, and the whole statement is false. We do not need to evaluate the second expression (birth_year <= 2005).

Similarly, for 

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bool is_not_gen_z(long birth_year)
{
  return ((birth_year < 1995) || (birth_year > 2005));
}

When birth_year is 1970, the expression (birth_year < 1995) is true, so we know that the whole statement is true. There is no need to check if (birth_year > 2005).

In both examples above, the savings due to short-circuiting is not much -- since we are comparing two numbers, and there is no side effects in comparing birth_year to 2005. But, let's suppose that we introduce two functions with side effects (of printing to the screen):

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bool not_too_old(long birth_year)
{
  if (birth_year >= 1995) {
    cs1010_print_string("not too old..");
    return true;
  }
  cs1010_print_string("too old..");
  return false;
}

bool not_too_young(long birth_year)
{
  if (birth_year <= 2005) {
    cs1010_print_string("not too young..");
    return true;
  }
  cs1010_print_string("too young..");
  return false;
}

bool is_gen_z(long birth_year)
{
  return not_too_old(birth_year) && not_too_young(birth_year);
}

When we call is_gen_z(1984), you might expect too old..not too young.. to be printed, but due to short-circuiting, the code only prints too old...

Another reason to keep short-circuiting in mind is that the order of the logical expressions matter: we would want to put the logical expression that involves more work in the second half of the expression. Take the following example:

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if (number < 100000 && is_prime(number)) {
    :
}

Checking whether a number is below 100,000 is easier than checking if a number is prime. So, we can skip checking for primality if the number is too big. Compare this to:

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if (is_prime(number) && number < 100000) {
    :
}

Suppose number is a gigantic integer, then we would have spent lots of effort checking if number is a prime, only to find out that it is too big anyway!

Problem Sets

Problem 9.1

Consider the function below, which aims to return the maximum value given three numbers.

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long max_of_three(long a, long b, long c)
{
  long max = 0;
  if ((a > b) && (a > c)) {
    // a is larger than b and c
    max = a;
  }
  if ((b > a) && (b > c)) {
    // b is larger than a and c
    max = b;
  }
  if ((c > a) && (c > b)) {
    // c is larger than a and b
    max = c;
  }
  return max;
}

(a) The function is not correct. Give a sample test value of a, b, and c that would expose the bug.

(b) List all conditions on the inputs a, b, and c such that the function above would fail.

Problem 9.2

The restaurant WcDonald's is setting a new rule for dining in. Two people are allowed to dine in only if both of them are fully vaccinated against COVID-19. A child below 12 years old from the same household is exempted from the rule.

Suppose we represent each diner with a long variable, and we have the following functions:

  • bool is_vaccinated(long p) returns true if and only if p is fully vaccinated.
  • bool is_a_child(long p) returns true if and only if p is a child below 12.
  • bool are_from_same_household(long p, long q) returns true if and only if both p and q are from the same household.

(a) Add as many rows as needed to the table below to enumerate all conditions under which p and q can dine in together at WcDonald's. Each table cell can be yes, no, or don't care.

is_vaccinated(p) is_vaccinated(q) is_a_child(p) is_a_child(q) are_from_same_household(p,q)
yes yes don't care don't care don't care
? ? ? ? ?

The first row has been filled up for you. It represents the condition in which both diners are fully vaccinated. In this case, it does not matter whether they are from the same household or they are children.

(b) Using the table from (a) to help you write a function can_dine_in(long p, long q) that returns true if and only if p and q can dine in together at WcDonald's.