This is for a computer database class, thank you! Prove or disprove the followin

Question

This is for a computer database class, thank you!

Prove or disprove the following inference rules for functional dependencies. A proof can be made either by a proof argument or by using inference rules IR1 through IR3. A disproof should be done by demonstrating a relation instance that satisfies the conditions and functional dependencies in the left hand side of the inference rule but do not satisfy the conditions or dependencies in the right hand side. {W rightarrow Y, X rightarrow Z} | = {WX rightarrow Y } {X rightarrow Y} and Z subset-of Y | = {X rightarrow Z} { X rightarrow Y, X rightarrow W, WY rightarrow Z} | = {X rightarrow Z}

Explanation / Answer

(a) Proof:
(1) W ->Y (given)
(2) X ->Z (given)
(3) WX ->YX (using IR2 (augment
ation) to augment (1) with X)
(4) YX ->Y (using IR1 (reflexivity)
, knowing that Y subset-of YX)
(5) WX ->Y (using IR3 (tr
ansitivity) on (3) and (4))
(b) Proof:
(1) X ->Y (given)
(2) Y ->Z (using IR1 (reflexivity)
, given that Z subset-of Y)
(3) X ->Z (using IR3 (transitivity) on (1) and (2))
(c) Proof:
(1) X ->Y (given)
(2) X ->W (given)
(3) WY ->Z (given)
(4) X ->XY (using IR2 (augmentat
ion) to augment (1) with X)
(5) XY ->WY (using IR2 (augmentat
ion) to augment (2) with Y)
(6) X ->WY (using IR3 (transitivity) on (4) and (5))
(7) X ->Z (using IR3 (transitivity) on (6) and (3))
(d) Disproof: X Y Z W
t 1 =x 1 y 1 z 1 w 1
t 2 =x 1 y 2 z 2 w 1
The above two tuples satisfy XY ->Z
and Y ->W but do not satisfy XW ->Z
(e) Disproof: X Y Z
t 1 =x 1 y 1 z 1
t 2 =x 1 y 2 z 1
The above two tuples satisfy X ->Z
and Y ->Z but do not satisfy X ->Y
(f) Proof:
(1) X ->Y (given)
(2) XY ->Z (given)
(3) X ->XY (using IR2 (augmentat
ion) to augment (1) with X)
(4) X ->Z (using IR3 (transitivity) on (3) and (2))

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.