Why does IQueryable.All() return true on an empty collection?

by

If my driveway is empty, I cannot assert that all cars parked there are red.

Consider the following statements.

S1: My driveway is empty.

S2: All the cars parked in my driveway are red.

I claim that S1 implies S2. That is, the statement S1 => S2 is true. I will do this by showing that its negation is false. In this case, the negation of S1 => S2 is S1 ^ ~S2; this is because S1 => S2 is false only when S1 is true and S2 is false. What is the negation of S2? It is

~S2: There exists a car parked in my driveway that is not red.

What is the truth value of S1 ^ ~S2? Let’s write it out

S1 ^ ~S2: My driveway is empty and there exists a car parked in my driveway that is not red.

The only way that S1 ^ ~S2 can be true is if both S1 and ~S2 are true. But S1 says that my driveway is empty and S2 says that there exists a car in my driveway. My driveway can not be both empty and contain a car. Thus, it is impossible for S1 and ~S2 to both be true. Therefore, S1 ^ ~S2 is false so its negation S1 => S2 is true.

Therefore, if your driveway is empty you can assert that all cars parked there are red.

So now let’s consider an IEnumerable<T> elements and a Predicate<T> p. Let us suppose that elements is empty. We wish to discover the value of

bool b = elements.All(x => p(x));

Let’s consider its negation

bool notb = elements.Any(x => !p(x));

For notb to be true, there must be at least one x in elements for which !p(x) is true. But elements is empty so it is impossible to find an x for which !p(x) is true. Therefore notb can not be true so it must be false. Since notb is false, its negation is true. Therefore b is true and elements.All(x => p(x)) must be true if elements is empty.

Here’s one more way to think of this. The predicate p is true if for all x in elements you can not find any for which it is false. But if there are no items in elements then it is impossible to find any for which it is false. Thus, for an empty collection elements, p is true for all x in elements

Now, what about elements.Any(x => p(x)) when elements is an empty IEnumerable<T> and p is a Predicate<T> as above? We already know the result will be false because we know its negation is true, but let’s reason through it anyway; the intuition is valuable. For elements.Any(x => p(x)) to be true there must be at least one x in elements for which p(x) is true. But if there aren’t any x in elements it is impossible to find any x for which p(x) is true. Therefore, elements.Any(x => p(x)) is false if elements is empty.

Finally, here’s a related explanation on why s.StartsWith(String.Empty) is true when s is a non-null instance of string:

More Related Contents:

  1. LINQ to Entities does not recognize the method
  2. Which method performs better: .Any() vs .Count() > 0?
  3. What is the difference between IQueryable and IEnumerable?
  4. LINQ: When to use SingleOrDefault vs. FirstOrDefault() with filtering criteria
  5. How to do SQL Like % in Linq?
  6. Is it better to call ToList() or ToArray() in LINQ queries?
  7. Linq to Entities – SQL “IN” clause
  8. Case insensitive string compare in LINQ-to-SQL
  9. LINQ on the .NET 2.0 Runtime
  10. Wrap a delegate in an IEqualityComparer
  11. “Or” equivalent in Linq Where() lambda expression
  12. How do I get the MAX row with a GROUP BY in LINQ query?
  13. LINQ Single vs First
  14. Linq int to string
  15. EF LINQ include multiple and nested entities
  16. .NET List Concat vs AddRange
  17. .NET LINQ query syntax vs method chain
  18. Ignore namespaces in LINQ to XML
  19. Does LINQ work with IEnumerable?
  20. LINQ, can’t join to string
  21. How to use “contains” or “like” in a dynamic linq query?
  22. Entity Framework 4 Single() vs First() vs FirstOrDefault()
  23. How to Convert LINQ Comprehension Query Syntax to Method Syntax using Lambda
  24. What can I do to resolve a “Row not found or changed” Exception in LINQ to SQL on a SQL Server Compact Edition Database?
  25. Why are some object properties UnaryExpression and others MemberExpression?
  26. How to perform merge sort using LINQ?
  27. Linq: What is the difference between Select and Where
  28. Linq to Objects: does GroupBy preserve order of elements?
  29. How to Convert Row to Column in Linq and SQL
  30. How to return a page of results from SQL?

Leave a Comment