A ⊆ B <===> A = (A ∩ B) 
We first prove: A ⊆ B ===> A = (A ∩ B) Given: A ⊆ B To prove that 2 set are equal: A = (A ∩ B) we must prove that: A ⊆ (A ∩ B) and (A ∩ B) ⊆ A First, this is trivially true: (A ∩ B) ⊆ A All that remains is to prove: A ⊆ (A ∩ B) Let x be an arbitrary element in A Then: x ∈ A And also: x ∈ B, because A ⊆ B (given) Therefore: x ∈ (A ∩ B) So: ∀ x ∈ A => x ∈ (A ∩ B) I.e.: A ⊆ (A ∩ B) 

Example:
A = { 1, 2, 3 } size(A) = 3 B = { 1, 4, 6 } size(B) = 3 
Size(A) = Size(B), but the sets A and B are not equal to each other.

Illustration:


SELECT fname, lname FROM employee WHERE "set of projects worked on by emplyee.ssn" CONTAINS "set of projects controlled by department 4" 
SELECT fname, lname FROM employee WHERE "set of projects worked on by emplyee.ssn" CONTAINS "set of projects controlled by department 4" 

By the property discussed above, we test the equality of sets by testing the size of these 2 sets:
SELECT fname, lname FROM employee WHERE "set of projects worked on by emplyee.ssn AND controlled by dept 4" = "set of projects controlled by department 4" 

SELECT pname FROM project P WHERE "set of employees working on project P" ⊆ "set of employees in the 'Research' department" 
SELECT pname FROM project P WHERE "set of employees working on project P" ⊆ "set of employees in the 'Research' department" 