6.7 — Logical operators

While relational (comparison) operators can be used to test whether a particular condition is true or false, they can only test one condition at a time. Often we need to know whether multiple conditions are true simultaneously. For example, to check whether we’ve won the lottery, we have to compare whether all of the multiple numbers we picked match the winning numbers. In a lottery with 6 numbers, this would involve 6 comparisons, all of which have to be true. In other cases, we need to know whether any one of multiple conditions is true. For example, we may decide to skip work today if we’re sick, or if we’re too tired, or if we won the lottery in our previous example. This would involve checking whether any of 3 comparisons is true.

Logical operators provide us with the capability to test multiple conditions.

C++ has 3 logical operators:

Logical NOT

You have already run across the logical NOT unary operator in lesson 4.9 -- Boolean values. We can summarize the effects of logical NOT like so:

If logical NOT’s operand evaluates to true, logical NOT evaluates to false. If logical NOT’s operand evaluates to false, logical NOT evaluates to true. In other words, logical NOT flips a Boolean value from true to false, and vice-versa.

Logical NOT is often used in conditionals:

bool tooLarge { x > 100 }; // tooLarge is true if x > 100
if (!tooLarge)
    // do something with x
else
    // print an error

One thing to be wary of is that logical NOT has a very high level of precedence. New programmers often make the following mistake:

#include <iostream>

int main()
{
    int x{ 5 };
    int y{ 7 };

    if (!x > y)
        std::cout << x << " is not greater than " << y << '\n';
    else
        std::cout << x << " is greater than " << y << '\n';

    return 0;
}

This program prints:

5 is greater than 7

But x is not greater than y, so how is this possible? The answer is that because the logical NOT operator has higher precedence than the greater than operator, the expression ! x > y actually evaluates as (!x) > y. Since x is 5, !x evaluates to 0, and 0 > y is false, so the else statement executes!

The correct way to write the above snippet is:

#include <iostream>

int main()
{
    int x{ 5 };
    int y{ 7 };

    if (!(x > y))
        std::cout << x << " is not greater than " << y << '\n';
    else
        std::cout << x << " is greater than " << y << '\n';

    return 0;
}

This way, x > y will be evaluated first, and then logical NOT will flip the Boolean result.

Simple uses of logical NOT, such as if (!value) do not need parentheses because precedence does not come into play.

Logical OR

The logical OR operator is used to test whether either of two conditions is true. If the left operand evaluates to true, or the right operand evaluates to true, or both are true, then the logical OR operator returns true. Otherwise it will return false.

For example, consider the following program:

#include <iostream>

int main()
{
    std::cout << "Enter a number: ";
    int value {};
    std::cin >> value;

    if (value == 0 || value == 1)
        std::cout << "You picked 0 or 1\n";
    else
        std::cout << "You did not pick 0 or 1\n";
    return 0;
}

In this case, we use the logical OR operator to test whether either the left condition (value == 0) or the right condition (value == 1) is true. If either (or both) are true, the logical OR operator evaluates to true, which means the if statement executes. If neither are true, the logical OR operator evaluates to false, which means the else statement executes.

You can string together many logical OR statements:

if (value == 0 || value == 1 || value == 2 || value == 3)
     std::cout << "You picked 0, 1, 2, or 3\n";

New programmers sometimes confuse the logical OR operator (||) with the bitwise OR operator (|) (covered in lesson O.2 -- Bitwise operators). Even though they both have OR in the name, they perform different functions. Mixing them up will probably lead to incorrect results.

Logical AND

The logical AND operator is used to test whether both operands are true. If both operands are true, logical AND returns true. Otherwise, it returns false.

For example, we might want to know if the value of variable x is between 10 and 20. This is actually two conditions: we need to know if x is greater than 10, and also whether x is less than 20.

#include <iostream>

int main()
{
    std::cout << "Enter a number: ";
    int value {};
    std::cin >> value;

    if (value > 10 && value < 20)
        std::cout << "Your value is between 10 and 20\n";
    else
        std::cout << "Your value is not between 10 and 20\n";
    return 0;
}

In this case, we use the logical AND operator to test whether the left condition (value > 10) AND the right condition (value < 20) are both true. If both are true, the logical AND operator evaluates to true, and the if statement executes. If neither are true, or only one is true, the logical AND operator evaluates to false, and the else statement executes.

As with logical OR, you can string together many logical AND statements:

if (value > 10 && value < 20 && value != 16)
    // do something
else
    // do something else

If all of these conditions are true, the if statement will execute. If any of these conditions are false, the else statement will execute.

As with logical and bitwise OR, new programmers sometimes confuse the logical AND operator (&&) with the bitwise AND operator (&).

Short circuit evaluation

In order for logical AND to return true, both operands must evaluate to true. If the left operand evaluates to false, logical AND knows it must return false regardless of whether the right operand evaluates to true or false. In this case, the logical AND operator will go ahead and return false immediately without even evaluating the right operand! This is known as short circuit evaluation, and it is done primarily for optimization purposes.

Similarly, if the left operand for logical OR is true, then the entire OR condition has to evaluate to true, and the right operand won’t be evaluated.

Short circuit evaluation presents another opportunity to show why operators that cause side effects should not be used in compound expressions. Consider the following snippet:

if (x == 1 && ++y == 2)
    // do something

if x does not equal 1, the whole condition must be false, so ++y never gets evaluated! Thus, y will only be incremented if x evaluates to 1, which is probably not what the programmer intended!

Mixing ANDs and ORs

Mixing logical AND and logical OR operators in the same expression often can not be avoided, but it is an area full of potential dangers.

Because logical AND and logical OR seem like a pair, many programmers assume they have the same precedence (just like addition/subtraction and multiplication/division). However, logical AND has higher precedence than logical OR, thus logical AND operators will be evaluated ahead of logical OR operators (unless they have been parenthesized).

New programmers will often write expressions such as value1 || value2 && value3. Because logical AND has higher precedence, this evaluates as value1 || (value2 && value3), not (value1 || value2) && value3. Hopefully that’s what the programmer wanted! If the programmer was assuming left to right association (as happens with addition/subtraction, or multiplication/division), the programmer will get a result he or she was not expecting!

When mixing logical AND and logical OR in the same expression, it is a good idea to explicitly parenthesize each operator and its operands. This helps prevent precedence mistakes, makes your code easier to read, and clearly defines how you intended the expression to evaluate. For example, rather than writing value1 && value2 || value3 && value4, it is better to write (value1 && value2) || (value3 && value4).

De Morgan’s laws

Many programmers also make the mistake of thinking that !(x && y) is the same thing as !x && !y. Unfortunately, you can not “distribute” the logical NOT in that manner.

De Morgan’s laws tell us how the logical NOT should be distributed in these cases:

!(x && y) is equivalent to !x || !y
!(x || y) is equivalent to !x && !y

In other words, when you distribute the logical NOT, you also need to flip logical AND to logical OR, and vice-versa!

This can sometimes be useful when trying to make complex expressions easier to read.

Where’s the logical exclusive or (XOR) operator?

Logical XOR is a logical operator provided in some languages that is used to test whether an odd number of conditions is true.

C++ doesn’t provide a logical XOR operator (operator^ is a bitwise XOR, not a logical XOR). Unlike logical OR or logical AND, logical XOR cannot be short circuit evaluated. Because of this, making a logical XOR operator out of logical OR and logical AND operators is challenging. However, you can easily mimic logical XOR using the inequality operator (!=):

if (a != b) ... // a XOR b, assuming a and b are Booleans

This can be extended to multiple operands as follows:

if (a != b != c != d) ... // a XOR b XOR c XOR d, assuming a, b, c, and d are Booleans

Note that the above XOR patterns only work if the operands are Booleans (not integers).

if (static_cast<bool>(a) != static_cast<bool>(b) != static_cast<bool>(c) != static_cast<bool>(d)) ... // a XOR b XOR c XOR d, for any type that can be converted to bool

if (!!a != !!b != !!c != !!d)

Alternative operator representations

Many operators in C++ (such as operator ||) have names that are just symbols. Historically, not all keyboards and language standards have supported all of the symbols needed to type these operators. As such, C++ supports an alternative set of keywords for the operators that use words instead of symbols. For example, instead of ||, you can use the keyword or.

The full list can be found here. Of particular note are the following three:

This means the following are identical:

std::cout << !a && (b || c);
std::cout << not a and (b or c);

While these alternative names might seem easier to understand right now, most experienced C++ developers prefer using the symbolic names over the keyword names. As such, we recommend learning and using the symbolic names, as this is what you will commonly find in existing code.

Quiz time

Question #1

Evaluate the following expressions.

Note: in the following answers, we “explain our work” by showing you the steps taken to get to the final answer. The steps are separated by a => symbol. Expressions that were ignored due to the short circuit rules are placed in square brackets. For example
(1 < 2 || 3 != 3) =>
(true || [3 != 3]) =>
(true) =>
true
means we evaluated (1 < 2 || 3 != 3) to arrive at (true || [3 != 3]) and evaluated that to arrive at “true”. The 3 != 3 was never executed due to short circuiting.

a) (true && true) || false

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b) (false && true) || true

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c) (false && true) || false || true

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d) (5 > 6 || 4 > 3) && (7 > 8)

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e) !(7 > 6 || 3 > 4)

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Question #2

In lesson 6.3 -- Remainder and Exponentiation, we wrote a function to determine whether a number is even that looked like this:

#include <iostream>

bool isEven(int x)
{
    // if x % 2 == 0, 2 divides evenly into our number, which means it must be an even number
    return (x % 2) == 0;
}

int main()
{
    std::cout << "Enter an integer: ";
    int x{};
    std::cin >> x;

    if (isEven(x))
        std::cout << x << " is even\n";
    else
        std::cout << x << " is odd\n";

    return 0;
}

Rewrite this function using operator! instead of operator==.

Show Solution

#include <iostream>

bool isEven(int x)
{
    return !(x % 2);
}

int main()
{
    std::cout << "Enter an integer: ";
    int x{};
    std::cin >> x;

    if (isEven(x))
        std::cout << x << " is even\n";
    else
        std::cout << x << " is odd\n";

    return 0;
}