Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider the relation R(X, Y, Z, W, V) in whi...
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Consider the relation R(X, Y, Z, W, V) in which X, Y, Z, W, and V are the attributes and the following set of functional dependencies
A = {{X, Y} → {Z, W}, {X, W, V} → {Y, Z}}
Which of the following is the trivial functional dependency in A+, where A+ is closure of A?
A = {{X, Y} → {Z, W}, {X, W, V} → {Y, Z}}
Which of the following is the trivial functional dependency in A+, where A+ is closure of A?
- a){X, Z} → {Z, W}
- b){X, V} → {V}
- c){X, W, V} → {Y, W}
- d){Y, W} → {Y, X}
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Consider the relation R(X, Y, Z, W, V) in which X, Y, Z, W, and V are ...
Concept:
The closure of F, denoted as F+, is the set of all regular FD, that can be derived from.
For trivial functional dependency,
Let A and be two sets consists of attributes of a relation
A → B
A ⊇ B
Option 1:
{X, Z} → {Z, W}
{X, Z} ⊉ {Z, W}
Not a trivial functional dependency
The closure of F, denoted as F+, is the set of all regular FD, that can be derived from.
For trivial functional dependency,
Let A and be two sets consists of attributes of a relation
A → B
A ⊇ B
Option 1:
{X, Z} → {Z, W}
{X, Z} ⊉ {Z, W}
Not a trivial functional dependency
Option 2:
{X, V} → {V}
{X, V} ⊇ {V}
It is a trivial functional dependency
{X, V} → {V}
{X, V} ⊇ {V}
It is a trivial functional dependency
Option 3:
{X, W, V} → {Y}
{X, W, V} ⊉ {Y}
Not a trivial functional dependency
{X, W, V} → {Y}
{X, W, V} ⊉ {Y}
Not a trivial functional dependency
Option 4:
{Y, W} → {Y, X}
{Y, W} ⊉ {Y, X}
Not a trivial functional dependency
NOTE:
⊇ → superset
⊉ → not superset
{Y, W} → {Y, X}
{Y, W} ⊉ {Y, X}
Not a trivial functional dependency
NOTE:
⊇ → superset
⊉ → not superset
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Community Answer
Consider the relation R(X, Y, Z, W, V) in which X, Y, Z, W, and V are ...
Z, {Y, Z} -> W, {W, V} -> X}
To determine the candidate keys for this relation, we can use the closure of attributes method.
Starting with the given functional dependencies, we can find the closure of each attribute set.
For X:
X+ = X
No other functional dependency has X on the right side, so X is not a candidate key.
For Y:
Y+ = Y
{X, Y} -> Z, so Y+ = YZ
{Y, Z} -> W, so Y+ = YZW
{W, V} -> X, so Y+ = YZWVX
Since Y+ contains all the attributes, Y is a candidate key.
For Z:
Z+ = Z
{X, Y} -> Z, so Z+ = Z
No other functional dependency has Z on the right side, so Z is not a candidate key.
For W:
W+ = W
{W, V} -> X, so W+ = WX
No other functional dependency has W on the right side, so W is not a candidate key.
For V:
V+ = V
{W, V} -> X, so V+ = VX
No other functional dependency has V on the right side, so V is not a candidate key.
Therefore, the only candidate key for this relation is Y.
To determine the candidate keys for this relation, we can use the closure of attributes method.
Starting with the given functional dependencies, we can find the closure of each attribute set.
For X:
X+ = X
No other functional dependency has X on the right side, so X is not a candidate key.
For Y:
Y+ = Y
{X, Y} -> Z, so Y+ = YZ
{Y, Z} -> W, so Y+ = YZW
{W, V} -> X, so Y+ = YZWVX
Since Y+ contains all the attributes, Y is a candidate key.
For Z:
Z+ = Z
{X, Y} -> Z, so Z+ = Z
No other functional dependency has Z on the right side, so Z is not a candidate key.
For W:
W+ = W
{W, V} -> X, so W+ = WX
No other functional dependency has W on the right side, so W is not a candidate key.
For V:
V+ = V
{W, V} -> X, so V+ = VX
No other functional dependency has V on the right side, so V is not a candidate key.
Therefore, the only candidate key for this relation is Y.
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