OPTION : (d)
The following query states the following conditions:
Sex = F $\land$
x= M $\land$
Marks $\leq$ m
Let the relation Student(Name, Sex, Marks)
Name

Sex

Marks

S1

F

30

S2

F

10

S3

M

20

Student(Name, Sex, Marks) Relation is renamed as Student(x, x, m).
Taking cross product of the relations
No

Name

Sex

Marks

n

x

m

1

S1

F

30

S1

F

30

2

S1

F

30

S2

F

10

3

S1

F

30

S3

M

20

4

S2

F

10

S1

F

30

5

S2

F

10

S2

F

10

6

S2

F

10

S3

M

20

7

S3

M

20

S1

F

30

8

S3

M

20

S2

F

10

9

S3

M

20

S3

M

20

Selecting the tuple (row# 6 from the above table), which satisfies the condition (i.e.,)
Sex = F $\land$
x = M $\land$
Marks $\leq$ m
PROJECTING:
$\Pi_{name} = S2$
$\Pi_{name} ( \sigma_{sex=F} (student) ) = \begin{matrix} S1 \\ S2 \end{matrix}$
Hence the query:
$\Pi_{name} \begin{bmatrix} \sigma_{sex=F} (student) \end{bmatrix}  \Pi_{name} \begin{bmatrix} student \bowtie \sigma_{x,x,m} (student) \\ sex = F \wedge \\ x= M \wedge \\ marks \leq m \end{bmatrix}$
$\begin{matrix} S1 \\ S2 \end{matrix} – S2 = S1$
Let us take another relation data of student(Name, Sex, marks)
Name

Sex

Marks


S1

M

100

> highest marks of M student

S2

F

50

> highest marks of F student

S3

M

40


S4

F

30


Taking the cross product
NO

Name

Sex

Marks

x

x

M

1

S1

M

100

S1

M

100

2

S1

M

100

S2

F

50

3

S1

M

100

S3

M

40

4

S1

M

100

S4

F

30

5

S2

F

50

S1

M

100

6

S2

F

50

S2

F

50

7

S2

F

50

S3

M

40

8

S2

F

50

S4

F

30

9

S3

M

100

S1

M

100

10

S3

M

100

S2

F

50

11

S3

M

100

S3

M

40

12

S3

M

100

S4

F

30

13

S4

F

50

S1

M

100

14

S4

F

50

S2

F

50

15

S4

F

50

S3

M

40

16

S4

F

50

S4

F

30

Consider the row numbers 5, 13, 15 from the above table,
$\begin{matrix} S2 \\ S4 \end{matrix}$ => F students who score less than equal to some M students.
$\Pi_{name} [ \sigma_{sex=F} (student) ] = \begin{matrix} S2 \\ S4 \end{matrix}$
Hence, the result of the query will be:
$\begin{matrix} S2 \\ S4 \end{matrix}  \begin{matrix} S2 \\ S4 \end{matrix} = empty relation
From the above relational data of table Student(Name, Sex, Marks)
(D) is the correct option
In short,
$\{ \geq \text{All} \} = \mid \text{universal}\mid  \mid < \text{some M} \mid$
$\{ > \text{All} \} = \mid \text{universal} \mid  \mid \leq \text{some M} \mid$
$\{ \geq some \} = \mid universal \mid  \mid < all M \mid$
Source : http://www.edugrabs.com/questionsbasedonrelationalalgebra/