# 1.10. Set operations¶

Relational database theory is based on mathematical set theory. Even though relational database implementations stray from the theory in some important regards, the notion of sets remains important. In this chapter, we examine the three set operations available to us in SQL.

## 1.10.1. Tables used in this chapter¶

For this chapter we will be using the books dataset (tables books, authors, etc.), which is described in Appendix A.

## 1.10.2. Sets refresher¶

If you are already familiar with sets, and with basic operations on sets (union, intersection, and set difference), then you can skip this section. Otherwise, please continue reading for some very basic background.

A set is a mathematical object that represents a collection of distinct values. For a given set and any value, we can ask whether or not the set contains the value. Sets can be defined by some property that values have in common, or simply by listing all of the values in the set. For example, in the figure below, the blue circle (including the overlapping portion) represents a set containing the numbers 0, 2, 4, 6, 8, and 10. More succinctly, this set contains the multiples of 2 between 0 and 10. The figure also contains a yellow circle containing the multiples of 3 between 0 and 10.

This type of diagram is known as a Venn diagram, and it is frequently used to illustrate sets and operations on them. We will use it to discuss three binary operations on sets: union, intersection, and set difference.

The union of two sets is another set: the set containing all values that are in either set. In the diagram above, the union of the two sets contains the values 0, 2, 3, 4, 6, 8, 9, and 10. In the diagram, this is every number that is contained in either circle. Note that we do not duplicate values; even though 6 is in both sets, the union of the sets does not contain 6 twice. A number is either in the set once, or it is not in the set at all. A union of values is related to the Boolean OR operator: the union of these two sets contains integers between 0 and 10 which are multiples of 2 OR multiples of 3.

The intersection of two sets is again a set, this time containing only values which appear in both of the original sets. In the diagram above, the intersection is represented by the overlap between the two circles, containing the values 0 and 6. Intersection is related to the Boolean AND operation; the intersection of these two sets contains integers between 0 and 10 which are multiples of 2 AND multiples of 3.

While union and intersection are commutative - the sets involved in an operation can be exchanged and get the same result - set difference is not. In set difference, you are “subtracting” one set from another to obtain a new set. The result is all values in the first set excluding any values also in the second set. The diagram above shows the two possible set differences we can obtain using our two sets. These are the portions of the circles that are outside the intersection. For example, if we subtract the set of multiples of 3 from the set of multiples of 2, we get the numbers in the left circle which are not in the right circle, that is, the values 2, 4, 8, and 10. Set difference does not correspond directly to a basic Boolean operation, but we can approximate set difference using NOT and AND: the set difference above (multiples of 2 minus multiples of 3) is the set of integers between 0 and 10 inclusive which are multiples of 2 AND NOT multiples of 3.

It may be difficult to imagine all of the amazing applications of sets in both mathematics and computer science from this simple example. However, set theory is a very powerful tool. As we will discuss in Part 3, relational databases resulted from the application of set theory to the problems of data management.

## 1.10.3. Tables as sets¶

Mathematically, sets are collections of distinct values. In the original conception of relational databases, tables and the results of data retrieval queries were intended to be true sets; that is, collections of rows, with no two rows being exactly the same. For performance reasons, SQL databases allow duplicate rows in both tables and in the results of queries. For example, if we run the following query, we get some duplicate rows.

SELECT publication_year FROM books;


The term used to describe tables and query results in SQL is multiset. A multiset is a collection of values from the same domain of values, but values can appear more than once in the set. This difference between relational databases in practice and in theory results in some complications, as we will soon see.

The three basic set operations that SQL supports are union, intersection, and set difference.

### 1.10.3.1. Union¶

Set union in SQL is an operation on two SELECT queries. The query is written as one SELECT query, followed by the keyword UNION, followed by another SELECT query. The two query results must be compatible in the sense that they must both return the same number of columns, and the columns should have compatible types. The union of the queries contains every distinct row that is returned from either query. As a very simple example, we can use a UNION query in place of a Boolean OR condition. Compare these two queries:

In this case, the queries return the same results. However, there is a subtle difference between them. When we use UNION, SQL treats it as a true set operation and returns a set of distinct rows - any duplicates are removed. To be completely equivalent, we should use the DISTINCT keyword in the second query.

There is no particular reason to choose a union query over the OR expression in this case; it is merely used for illustration. UNION may be a more preferable alternative in other scenarios, such as those involving complex conditional logic. As a simple example, consider providing a column labeling authors as “living”, “dead” (giving the date of death), or “unknown” (where the birth and death dates are unknown). We could do this with a CASE expression, or with a UNION of three queries (think of a union of the first two queries, then a union of the result with the third query):

SELECT name, 'living' AS status
FROM authors
WHERE death IS NULL AND birth IS NOT NULL
UNION
SELECT name, 'died ' || death
FROM authors
WHERE death IS NOT NULL AND birth IS NOT NULL
UNION
SELECT name, 'unknown'
FROM authors
WHERE birth IS NULL;

SELECT
name,
CASE
WHEN death IS NULL AND birth IS NOT NULL
THEN 'living'
WHEN death IS NOT NULL AND birth IS NOT NULL
THEN 'died ' || death
WHEN birth IS NULL
THEN 'unknown'
END AS status
FROM authors;


If you run the union query above, you will see that column names for the result of the whole query come from the first SELECT query when using set operations.

In some cases, UNION may be your only choice - such as when you are combining results from different tables. One example of this might occur when a company wishes to create an email list for everyone related to the company in some way: the company’s database might contain one table for employees, another for customers, and a third for vendors A union query would easily create one mailing list from these three tables, and eliminate duplicates (since, for example, employees might also be customers).

#### 1.10.3.1.1. Multiset complication¶

Used by itself, UNION results in the removal of all duplicates from the result set of the query. There may be occasions when this is not the desired behavior; if you wish to retain duplicate records (keeping all rows returned by either query), simply add the keyword ALL after UNION. The query below will result in duplicate records:

SELECT * FROM books WHERE title LIKE 'W%'
UNION ALL
SELECT * FROM books WHERE publication_year = 1995;


### 1.10.3.2. Intersection¶

Set intersection in SQL is accomplished by the keyword INTERSECT. The rules for using INTERSECT are the same as for using UNION, but its result contains only every distinct row that is contained in both query results:

SELECT * FROM books WHERE title LIKE 'W%'
INTERSECT
SELECT * FROM books WHERE publication_year = 1995;


This result is similar to that achieved by using an AND expression in the WHERE clause of a single query:

SELECT DISTINCT *
FROM books
WHERE title LIKE 'W%'
AND publication_year = 1995;


However, as with UNION, you can use INTERSECT to perform queries against multiple tables.

The SQL standard allows the keyword ALL after INTERSECT, but most databases (including SQLite) do not support this usage.

(Note for MySQL users: MySQL does not implement INTERSECT.)

### 1.10.3.3. Set difference¶

Set difference in SQL is accomplished by the keyword EXCEPT. The rules for using EXCEPT are again the same as for UNION and INTERSECT, but note that EXCEPT is not commutative - the order of the queries matters. Here is our same example again, using EXCEPT:

SELECT * FROM books WHERE title LIKE 'W%'
EXCEPT
SELECT * FROM books WHERE publication_year = 1995;


This result is similar to that achieved by requiring one condition AND NOT the other condition in the WHERE clause of a single query:

SELECT DISTINCT *
FROM books
WHERE title LIKE 'W%'
AND NOT publication_year = 1995;


However, as with UNION and INTERSECT, you can use EXCEPT to perform queries against multiple tables.

The SQL standard allows the keyword ALL after EXCEPT, but most databases (including SQLite) do not support this usage.

One application of the EXCEPT operator is determining if two query results are identical; if you take the set difference in both directions, your result should be empty if the two queries return the same distinct rows (there could be a difference in the counts of duplicate rows). An alternate approach is to see if the union and intersection of the two queries contain the same count of rows.

(Note for MySQL users: MySQL does not implement EXCEPT.)

(Note for Oracle users: Oracle uses the keyword MINUS rather than EXCEPT.)

### 1.10.3.4. Chaining operations¶

As we saw with UNION, it is possible to do more than one set operation in a single query. For queries just involving UNION, the order of queries does not matter, as UNION is both commutative and associative. The same is true for a query just involving INTERSECT. For queries involving EXCEPT, or queries mixing set operations, the situation is more complicated. EXCEPT is neither commutative nor associative. Queries that chain mixed operators do not behave the same in all databases, so be cautious when attempting this; some databases allow you to use parentheses to force the order in which you want operations to be performed.

## 1.10.4. Self-check exercises¶

This section contains some exercises using the books data set (reminder: you can get full descriptions of all tables in Appendix A). If you get stuck, click on the “Show answer” button below the exercise to see a correct answer. There are many ways to answer these questions; try to use a set operation to solve each.

Write a query to find all of the awards won by poet Allen Ginsberg, either as an author or for a book. Your output should have three columns: the name of the award, the year of the award, and what the award was won for - the book title for book awards, or “body of work” for author awards.

Write a query to find a list of awards that have been given for either books or an author’s body of work (i.e., the award(s) should show up in both authors_awards and books_awards).

Write a query to find a list of authors who won author awards but no book awards.