# DBMS | Tuple Relational Calculus

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 tupple 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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