GATE CSE 1987 | Question: 2n

Question

State whether the following statements are TRUE or FALSE:

A relation $r$ with schema $(X, Y)$ satisfies the function dependency $X \rightarrow Y$, The tuples $\langle 1, 2\rangle$ and $\langle 2, 2 \rangle$ can both be in $r$ simultaneously.

Comments

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Video Solution

Answer 1

True is answer.

$X\to Y$ says when $X$ repeats, $Y$ will be also repeat. Since, $X$ is not repeated, $Y$ may or may not repeat. 
$$\begin{array}{|c|c|} \hline \text {X} & \text {Y} \\\hline \text {1} &  \text{2 } \\\hline \text{2} & \text{2}  \\\hline \end{array}$$

Comments

A functional dependency of the form $X\rightarrow Y$ implies 

for all possible pair of tuples $t_1,t_2 (t_1[X]=t_2[X])\rightarrow (t_1[Y]=t_2[Y])$ should hold.

Here in this case, the L.H.S of the implication is always false and hence implication is always true and hence the result is YES.

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@Ayush Upadhyaya how LHS of the implication is false?

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@vizzard110

For all possible pair of tuples $t_1,t_2 \left(t_1[X]=t_2[X])\rightarrow (t_1[Y]=t_2[Y]\right)$ should hold.

$t_{1}[X] = 1$ and  $t_{2}[X] = 2$

$\therefore \left(t_1[X] \neq t_2[X]\right)$

Answer 2

Rules for Functional Dependency-

 

Rule-01:

 

A functional dependency X → Y will always hold if all the values of X are unique (different) irrespective of the values of Y.

 

Example-

 

Consider the following table-

 

A	B	C	D	E
5	4	3	2	2
8	5	3	2	1
1	9	3	3	5
4	7	3	3	8

 

The following functional dependencies will always hold since all the values of attribute ‘A’ are unique-

• A → B
• A → BC
• A → CD
• A → BCD
• A → DE
• A → BCDE

In general, we can say following functional dependency will always hold-

 

A → Any combination of attributes A, B, C, D, E

 

Similar will be the case for attributes B and E.

 

Rule-02:

 

A functional dependency X → Y will always hold if all the values of Y are same irrespective of the values of X.

 

Example-

Consider the following table-

 

A	B	C	D	E
5	4	3	2	2
8	5	3	2	1
1	9	3	3	5
4	7	3	3	8

 

The following functional dependencies will always hold since all the values of attribute ‘C’ are same-

• A → C
• AB → C
• ABDE → C
• DE → C
• AE → C

 

In general, we can say following functional dependency will always hold true-

 

Any combination of attributes A, B, C, D, E → C

 

Combining Rule-01 and Rule-02 we can say-

 

In general, a functional dependency α → β always holds-   If either all values of α are unique or if all values of β are same or both.

 

Rule-03:

 

For a functional dependency X → Y to hold, if two tuples in the table agree on the value of attribute X, then they must also agree on the value of attribute Y.

 

Rule-04:

 

For a functional dependency X → Y, violation will occur only when for two or more same values of X, the corresponding Y values are different.

 

Courtesy:Gatevidyalay

Answer 3

t1(x) = t2(x) → t1(y) = t2(y),  but

t1(x) != t2(x) → t1(y) may/may not be t2(y)