Tuple Relational Calculus (TRC) in DBMS

Tuple Relational Calculus is a non-procedural query language unlike relational algebra. Tuple Calculus provides only the description of the query but it does not provide the methods to solve it. Thus, it explains what to do but not how to do. 
In Tuple Calculus, a query is expressed as 

{t| P(t)}

where t = resulting tuples, 
P(t) = known as Predicate and these are the conditions that are used to fetch t 

Thus, it generates set of all tuples t, such that Predicate P(t) is true for t. 

P(t) may have various conditions logically combined with OR (∨), AND (∧), NOT(¬). 
It also uses quantifiers: 
∃ t ∈ r (Q(t)) = ”there exists” a tuple in t in relation r such that predicate Q(t) is true. 
∀ t ∈ r (Q(t)) = Q(t) is true “for all” tuples in relation r. 

Example:
Table-1: Customer

Customer name	Street	City
Saurabh	A7	Patiala
Mehak	B6	Jalandhar
Sumiti	D9	Ludhiana
Ria	A5	Patiala

Table-2: Branch

Branch name	Branch city
ABC	Patiala
DEF	Ludhiana
GHI	Jalandhar

Table-3: Account

Account number	Branch name	Balance
1111	ABC	50000
1112	DEF	10000
1113	GHI	9000
1114	ABC	7000

Table-4: Loan

Loan number	Branch name	Amount
L33	ABC	10000
L35	DEF	15000
L49	GHI	9000
L98	DEF	65000

Table-5: Borrower

Customer name	Loan number
Saurabh	L33
Mehak	L49
Ria	L98

Table-6: Depositor

Customer name	Account number
Saurabh	1111
Mehak	1113
Sumiti	1114

Queries-1: Find the loan number, branch, amount of loans of greater than or equal to 10000 amount.  

{t| t ∈ loan  ∧ t[amount]>=10000}

Resulting relation: 

Loan number	Branch name	Amount
L33	ABC	10000
L35	DEF	15000
L98	DEF	65000

In the above query, t[amount] is known as tuple variable. 

Queries-2: Find the loan number for each loan of an amount greater or equal to 10000.  

{t| ∃ s ∈ loan(t[loan number] = s[loan number]  
                   ∧ s[amount]>=10000)}

Resulting relation: 

Loan number
L33
L35
L98

Queries-3: Find the names of all customers who have a loan and an account at the bank.  

{t | ∃ s ∈ borrower( t[customer-name] = s[customer-name])   
     ∧  ∃ u ∈ depositor( t[customer-name] = u[customer-name])}

Resulting relation: 

Customer name
Saurabh
Mehak

Queries-4: Find the names of all customers having a loan at the “ABC” branch.  

{t | ∃ s ∈ borrower(t[customer-name] = s[customer-name]  
   ∧ ∃ u ∈  loan(u[branch-name] = “ABC” ∧ u[loan-number] = s[loan-number]))}

Resulting relation: 

Customer name
Saurabh

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