Solved Problems

 

Problem 

Given:

F = { A → BC, CD → E, B → D, E → A }

Tasks:

  1. Find A⁺
  2. Prove whether A → E
  3. Check if A is a candidate key

Solution:

Step 1: Start closure

A⁺ = {A}

Step 2: Apply A → BC

A⁺ = {A, B, C}

Step 3: Apply B → D

A⁺ = {A, B, C, D}

Step 4: Apply CD → E

A⁺ = {A, B, C, D, E}

Step 5: Apply E → A (already included)

 Final:

A⁺ = {A, B, C, D, E}

Answers:

  • A → E ✅ (since E ∈ A⁺)
  • A is a candidate key ✅

Problem 

Given:

F = { AB → C, C → D, D → E, E → B }

Tasks:

  1. Find (AB)⁺
  2. Check if AB → E
  3. Find all attributes determined by AB

Solution:

Start:

(AB)⁺ = {A, B}

Apply:

  • AB → C → {A, B, C}
  • C → D → {A, B, C, D}
  • D → E → {A, B, C, D, E}
  • E → B (already present)

Final:

(AB)⁺ = {A, B, C, D, E}

Answers:

  • AB → E ✅
  • AB determines all attributes

Problem 3

Given:

F = { A → B, B → C, AC → D, D → E, E → F }

❓ Tasks:

  1. Find A⁺
  2. Is AC → F valid?
  3. Is A → D valid?

✅ Solution:

Step 1: A⁺

A → B → C
A⁺ = {A, B, C}

Step 2: Use AC → D

Since A⁺ contains C:

AC → D ⇒ A → D

Now:

A⁺ = {A, B, C, D}

Step 3: D → E → F

A⁺ = {A, B, C, D, E, F}

✔ Answers:

  • A⁺ = all attributes
  • AC → F ✅
  • A → D ✅ (important inference!)

Problem 

 Given:

F = { A → B, C → D }

Tasks:

Check if:

AC → BD

❌ Solution:

  • No rule allows combining unrelated FDs like this directly

✔ Answer:

AC → BD ❌ NOT VALID

Problem 

Given:

R(A, B, C, D, E)
F = { A → BC, CD → E, B → D }

Tasks:

  1. Find A⁺
  2. Find candidate keys

Solution:

Step 1: A⁺

A → BC ⇒ {A, B, C}
B → D ⇒ {A, B, C, D}
CD → E ⇒ {A, B, C, D, E}

✔ A⁺:

{A, B, C, D, E}

✔ Candidate Key:

A is a candidate key

Problem 

Given:

F = { AB → C, C → A, BC → D, D → E }

Tasks:

  1. Find (AB)⁺
  2. Find (BC)⁺
  3. Identify candidate keys

Solution:

(AB)⁺:

AB → C ⇒ {A, B, C}
C → A (already)
BC → D ⇒ {A, B, C, D}
D → E ⇒ {A, B, C, D, E}

(BC)⁺:

BC → D ⇒ {B, C, D}
D → E ⇒ {B, C, D, E}
C → A ⇒ {A, B, C, D, E}

✔ Candidate Keys:

AB and BC

Problem 

Given:

F = { A → B, B → C, CD → E, E → F, F → G }

Tasks:

  1. Find A⁺
  2. Prove A → G
  3. Check if A → E

Solution:

A → B → C
A⁺ = {A, B, C}

Cannot apply:

CD → E (missing D)

✔ Final:

A⁺ = {A, B, C}

✔ Answers:

  • A → G ❌
  • A → E ❌


Problem 

 Given:

F = { A → B, AB → C, C → D, D → E }

Tasks:

  1. Find A⁺
  2. Is A → E valid?

Solution:

A → B ⇒ {A, B}
AB → C ⇒ {A, B, C}
C → D ⇒ {A, B, C, D}
D → E ⇒ {A, B, C, D, E}

✔ Final:

A⁺ = {A, B, C, D, E}

✔ Answer:

A → E ✅

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