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1. AVA = A is called:

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

(C) Idempotent law      (D) Complement law

Answer: C

Explanation

1. Commutative Laws:

For any two finite sets A and B;

(i) A U B = B U A

(ii) A ∩ B = B ∩ A

2. 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.

3. Idempotent Laws:

For any finite set A;

(i) A U A = A

(ii) A ∩ A = A

4. 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.

5. 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’

More laws of algebra of sets:

6. For any two finite sets A and B;

(i) A – B = A ∩ B’

(ii) B – A = B ∩ A’

(iii) A – B = A ⇔ A ∩ B = ∅

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:

If f(x)=x+1 and g(x)=x+3, then fog(x)=x+4 and gof(x)=x+4

fof(x)=x+2, fofof(x)=x+3 and fofofof(x)=x+4.=x+3+1=g+1

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:

Context-free languages are closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:

• the union {\displaystyle L\cup P} of L and P
• the reversal of L
• the concatenation {\displaystyle L\cdot P} of L and P
• the Kleene star {\displaystyle L^{*}} of L
• the image {\displaystyle \varphi (L)} of L under a homomorphism {\displaystyle \varphi }
• the image {\displaystyle \varphi ^{-1}(L)} of L under an inverse homomorphism {\displaystyle \varphi ^{-1}}
• the cyclic shift of L (the language {\displaystyle \{vu:uv\in L\}})

Context-free languages are not closed under complement, intersection, or difference. However, if L is a context-free language and D is a regular language then both their intersection {\displaystyle L\cap D} and their difference {\displaystyle L\setminus D} are context-free languages.

ref: https://en.wikipedia.org/wiki/Context-free_language

4. 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, which of the above sequences are graphic?

(A) (i) and (ii)   

(B) (iii) and (iv)

(C) (iii) and (ii) 

(D) (ii) and (iv)

Answer: B

Explanation:

Rest can’t be graphs as the number of vertices with odd degree in a graph should be even.

In list (i), number of vertices with odd degree is 3.

In list (ii), number of vertices with odd degree is 3.

In list (iii), number of vertices with odd degree is 2.

In list (iv), number of vertices with odd degree is 2.

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

Explanation:

Basically, a field is a thing where you can add, subtract, multiply and divide. It is a bit tricky to see that the first three examples (Z23, Z29, Z31) are indeed fields. In fact, ZpZp happens to be a field always when pp is prime, and this result follows from Fermat’s little theorem.

But let us look at the fourth example. Assume you can divide the elements by 11, then you have

a contradiction. (The latter equality holds because 33=033=0 modulo 3333.) A similar argument shows you that ZqZq cannot be a field if qq is any composite number.

6. An example of a binary number which is equal to its 2’s complement is:

(A) 1100                        (B) 1001

(C) 1000                        (D) 1111

Answer: C

Explanation:

Option a : 1100 and 1s compliment 0011 +1 = 0100

and apply to all only c is the option which gives answer.

7. When a tri-state logic device is in the third state, then:

(A) it draws low current          

(B) it does not draw any current

(C) it draws very high current

(D) it presents a low impedance

Answer: D

Explanation:

Tristate means that the pin is used: to output a high state (1) or output a low state (2) AND it can be directed to enter a high-z/high impedance state (3). The last state, high impedance, is designed to prevent other active devices from driving this line simultaneously–never connect two active outputs together. Things that are not tristated would be a dedicated input or output lines. Any bi-directional lines (both input and output) are always tristated: example data bus.

8. An example of a connective which is not associative is:

(A) AND             (B) OR

(C) EX-OR        (D) NAND

Answer: D

check in detail: https://en.wikipedia.org/wiki/Logical_connective

9. Essential hazards may occur in:

(A) Combinational logic circuits

(B) Synchronous sequential logic circuits

(C) Asynchronous sequential logic circuits working in the fundamental mode

(D) Asynchronous sequential logic circuits working in the pulse mode

Answer: C

10. The characteristic equation of a T flip-flop is:

(A) Q_n+1=TQ’_n + T’ Q_n              

(B) Q_n+1=T+Q_n

(C) Q_n+1=TQ_n                            

(D) Q_n+1= T’Q’_n

The symbols used have the usual meaning.

Answer: A

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