The expression: (∃t) ( Condition(t) ) evaluates to true if there is some (at least one) tuple t that satisfies the condition Condition(t) false otherwise 
Examples:
(∃t) ( Employee(t) and t.fname='John' and t.lname='Smith') This expression is true if there is an employee named "John Smith" 
(∀t) ( Condition(t) ) evaluates to evaluates to true if all tuples t satisfies the condition Condition(t) false otherwise 
Example:
(∀t) ( Employee(t) and t.salary > 40000 ) This expression is true if all employees earn more than $40000 
Example:

Solution 1: without using the existential quantifier
{ d.dno  Department(d) and d.dname='Research' } 
Solution 2: with the existential quantifier ∃
{ d.dno  Department(d) and ( ( ∃ t ) ( Department(t) and t.dname = 'Research' and t.dno = d.dno ) ) } 
Explanation:

