Number of Possible Super Keys in DBMS

A super key is a set of one or more attributes that uniquely identifies each record in a table. All primary/candidate keys are super keys, but not all super keys are candidate keys (candidate keys are minimal super keys with no redundant attributes).

Key Formulas

Examples with Single Candidate Key

Example 1: R{a1, a2, a3}, candidate key = {a1}. Super keys = 2⁽³¹⁾ = 4 {a1}, {a1,a2}, {a1,a3}, {a1,a2,a3}.

Example 2: R{a1...an}, candidate key = {a1,a2,a3}. Super keys = 2⁽ⁿ³⁾.

Examples with Multiple Candidate Keys

Example 3: R{a1...an}, candidate keys = {a1}, {a2} ?

= SK(a1) + SK(a2) - SK(a1,a2)
= 2^(n-1) + 2^(n-1) - 2^(n-2)

Example 4: R{a1...an}, candidate keys = {a1}, {a2,a3} ?

= SK(a1) + SK(a2,a3) - SK(a1,a2,a3)
= 2^(n-1) + 2^(n-2) - 2^(n-3)

Example 5: R{a1...an}, candidate keys = {a1,a2}, {a3,a4} ?

= SK(a1,a2) + SK(a3,a4) - SK(a1,a2,a3,a4)
= 2^(n-2) + 2^(n-2) - 2^(n-4)

Example 6: R{a1...an}, candidate keys = {a1,a2}, {a1,a3} (overlapping) ?

= SK(a1,a2) + SK(a1,a3) - SK(a1,a2,a3)
= 2^(n-2) + 2^(n-2) - 2^(n-3)

Example 7: R{a1...an}, candidate keys = {a1}, {a2}, {a3} ?

Using Inclusion-Exclusion:
= 3 * 2^(n-1) - 3 * 2^(n-2) + 2^(n-3)

Practical Example

Example 8: R(A,B,C,D,E,F,G,H) with FDs: CH→G, A→BC, B→CFH, E→A, F→EG. The candidate keys are AD, BD, ED, FD. Applying inclusion-exclusion across all four candidate keys gives 120 super keys.

Maximum Candidate Keys

If any k attributes determine all others, the number of candidate keys is ⁿC_k. This is maximized when k = ⌊n/2⌋.

Conclusion

The number of super keys depends on the number of attributes and candidate keys. For a single candidate key with k attributes, use 2^(nk). For multiple candidate keys, apply the Inclusion-Exclusion principle to avoid double-counting overlapping super keys.