# Null Values

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Null values present special problems in relational operations, including arithmetic operations, comparison operations, and set operations.

The result of an arithmetic expression (involving, for example +, −, ∗, or /) is
null if any of the input values is null. For example, if a query has an expression
r.A+ 5, and r.A is null for a particular tuple, then the expression result must also
be null for that tuple.

Comparisons involving nulls are more of a problem. For example, consider
the comparison “1 < null”. It would be wrong to say this is true since we do not
know what the null value represents. But it would likewise be wrong to claim this
expression is false; if we did, “not (1 < null)” would evaluate to true, which does
not make sense. SQL therefore treats as unknown the result of any comparison
involving a null value (other than predicates is null and is not null, which are
described later in this section). This creates a third logical value in addition to true
and false.

Since the predicate in a where clause can involve Boolean operations such
as and, or, and not on the results of comparisons, the definitions of the Boolean
operations are extended to deal with the value unknown.

and: The result of true and unknown is unknown, false and unknown is false,
while unknown and unknown is unknown.

or: The result of true or unknown is true, false or unknown is unknown, while
unknown or unknown is unknown.

not: The result of not unknown is unknown.

You can verify that if r.A is null, then “1 < r.A” as well as “not (1 < r.A)”
evaluate to unknown.

If the where clause predicate evaluates to either false or unknown for a tuple,
that tuple is not added to the result.

SQL uses the special keyword null in a predicate to test for a null value. Thus,
to find all instructors who appear in the instructor relation with null values for
salary, we write:

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select name
from instructor
where salary is null;

The predicate is not null succeeds if the value on which it is applied is not null.

Some implementations of SQL also allow us to test whether the result of a com?parison is unknown, rather than true or false, by using the clauses is unknown
and is not unknown.

When a query uses the select distinct clause, duplicate tuples must be eliminated. For this purpose, when comparing values of corresponding attributes
from two tuples, the values are treated as identical if either both are non-null and
equal in value, or both are null. Thus two copies of a tuple, such as {(’A’,null),
(’A’,null)}, are treated as being identical, even if some of the attributes have a
null value. Using the distinct clause then retains only one copy of such identical
tuples. Note that the treatment of null above is different from the way nulls are
treated in predicates, where a comparison “null=null” would return unknown,
rather than true.

The above approach of treating tuples as identical if they have the same
values for all attributes, even if some of the values are null, is also used for the
set operations union, intersection and except.

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