UGC-NET Computer Science December 2004 Answer Key with Explanation

In this Blog we are providing all the UGC-NET Computer Science previous year Questions with explanation:

Q:: 1    A U A = A is called:

(A) Identity law             (B) De Morgan’s law

(C) Idempotent law       (D) Complement law

Answer: C

Explanation:

           (a) Commutative Laws:

                   For any two finite sets A and B;

                   (i) A U B = B U A

                   (ii) A ∩ B = B ∩ A

                   (b) Associative Laws:

                   For any three finite sets A, B and C;

                   (i) (A U B) U C = A U (B U C)

                   (ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

                   Thus, union and intersection are associative.

                   (c) Idempotent Laws:

                    For any finite set A;

                    (i) A U A = A

                    (ii) A ∩ A = A

                    (d) Distributive Laws:

                    For any three finite sets A, B and C;

                    (i) A U (B ∩ C) = (A U B) ∩ (A U C)

                    (ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

                    Thus, union and intersection are distributive over intersection and union respectively.

                    (e) De Morgan’s Laws:

                    For any two finite sets A and B;

                    (i) A – (B U C) = (A – B) ∩ (A – C)

                    (ii) A - (B ∩ C) = (A – B) U (A – C)

                    De Morgan’s Laws can also we written as:

                    (i) (A U B)’ = A' ∩ B'

                    (ii) (A ∩ B)’ = A' U B'

                    (f) For any two finite sets A and B;

                    (i) A – B = A ∩ B'

                    (ii) B – A = B ∩ A'

                    (iii) A – B = A ⇔ A ∩ B = ∅

                    (iv) (A – B) U B = A U B

                    (v) (A – B) ∩ B = ∅

                    (vi) A ⊆ B ⇔ B' ⊆ A'

                    (vii) (A – B) U (B – A) = (A U B) – (A ∩ B)

                    (g) For any three finite sets A, B and C;

                    (i) A – (B ∩ C) = (A – B) U (A – C)

                    (ii) A – (B U C) = (A – B) ∩ (A – C)

                    (iii) A ∩ (B - C) = (A ∩ B) - (A ∩ C)

                    (iv) A ∩ (B △ C) = (A ∩ B) △ (A ∩ C)



Q:: 2  If f(x) =x+1 and g(x)=x+3 then fofofof is:

(A) g       (B) g+1

(C) g⁴     (D) None of the above

        Answer: B

Explanation:

            fofofof is f(f(f(x+1))) i.e. f(f(x+2)) i.e. f(x+3) similarly it is equal to "x+4".

            straight away it is clear that it is g(x)+1 i.e. (x+3)+1 = x+4.
            it could also be like : fog(x)=x+4 and gof(x)=x+4.

Q:: 3  The context-free languages are closed for:

(i) Intersection                          (ii) Union

(iii) Complementation              (iv) Kleene Star

then

(A) (i) and (iv)               (B) (i) and (iii)

(C) (ii) and (iv)              (D) (ii) and (iii)

Answer: C

Explanation:

             Closure properties of languages::

                      This entry lists some common closure properties on the families of languages corresponding to the Chomsky hierarchy, as well as other related families.

operation	REG	DCFL	CFL	CSL	RC	RE
union	Y	N	Y	Y	Y	Y
intersection	Y	N	N	Y	Y	Y
set difference	Y	N	N	Y	Y	N
complementation	Y	Y	N	Y	Y	N
intersection with a regular language	Y	Y	Y	Y	Y	Y
concatenation	Y	N	Y	Y	Y	Y
Kleene star	Y	N	Y	Y	Y	Y
Kleene plus	Y	N	Y	Y	Y	Y
reversal	Y	Y	Y	Y	Y	Y
lambda-free homomorphism	Y	N	Y	Y	Y	Y
homomorphism	Y	N	Y	N	N	Y
inverse homomorphism	Y	Y	Y	Y	Y	Y
lambda-free substitution	Y	N	Y	Y	Y	Y
substitution	Y	N	Y	N	N	Y
lambda-free GSM mapping	Y	N	Y	Y	Y	Y
GSM mapping	Y	N	Y	N	N	Y
inverse GSM mapping	Y	Y	Y	Y	Y	Y
lambda- limited erasing	Y		Y	Y		Y
rational transduction	Y	N	Y	N	N	Y
right quotient with a regular language	Y	Y	Y	N		Y
left quotient with a regular language	Y	Y	Y	N		Y

                 
                where the definitions of the cells in the top row are the following language families:

Abbreviation	Name
REG	regular
DCFL	deterministic context-free
CFL	context-free
CSL	context-sensitive
RC	recursive
RE	recursively enumerable

Q:: 4  Which of the following lists are the degrees of all the vertices of a graph:

(i) 1, 2, 3, 4, 5                (ii) 3, 4, 5, 6, 7

(iii) 1, 4, 5, 8, 6              (iv) 3, 4, 5, 6

then

(A) (i) and (ii)   

(B) (iii) and (iv)

(C) (iii) and (ii) 

(D) (ii) and (iv)

Answer: B

Explanation::

           Sum of degrees of the vertices of a graph should be even, So, only option (iii) and (iv) satisfy this.

           Sum of degrees of the vertices of a graph is equal to twice the number of edges.

Q:: 5   If I_m denotes the set of integers modulo m, then the following are fields with respect to the operations of addition modulo m and multiplication modulo m:

         (i) Z₂₃                    (ii) Z₂₉

         (iii) Z₃₁                  (iv) Z₃₃

         Then

         (A) (i) only                                

         (B) (i) and (ii) only

         (C) (i), (ii) and (iii) only           

         (D) (i), (ii), (iii) and (iv)

         Answer: C

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