# Automata Theory Questions and Answers – The Language of DFA

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This set of Automata Theory Multiple Choice Questions & Answers (MCQs) focuses on “The Language of DFA”

1. How many languages are over the alphabet R?
a) countably infinite
b) countably finite
c) uncountable finite
d) uncountable infinite

Explanation: A language over an alphabet R is a set of strings over A which is uncountable and infinite.

2. According to the 5-tuple representation i.e. FA= {Q, ∑, δ, q, F}
Statement 1: q ϵ Q’; Statement 2: FϵQ
a) Statement 1 is true, Statement 2 is false
b) Statement 1 is false, Statement 2 is true
c) Statement 1 is false, Statement 2 may be true
d) Statement 1 may be true, Statement 2 is false

Explanation: Q is the Finite set of states, whose elements i.e. the states constitute the finite automata.

3. δˆ tells us the best:
a) how the DFA S behaves on a word u
b) the state is the dumping state
c) the final state has been reached
d) Kleene operation is performed on the set

Explanation: δ or the Transition function describes the best, how a DFA behaves on a string where to transit next, which direction to take.
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4. Which of the following option is correct?
A = {{abc, aaba}. {ε, a, bb}}
a) abcbb ₵ A
b) ε₵A
c) ε may not belong to A
d) abca ₵ A

Explanation: As the question has dot operation, ε will not be a part of the concatenated set. Had it been a union operation, ε would be a part of the operated set.

5. For a DFA accepting binary numbers whose decimal equivalent is divisible by 4, what are all the possible remainders?
a) 0
b) 0,2
c) 0,2,4
d) 0,1,2,3

Explanation: All the decimal numbers on division would lead to only 4 remainders i.e. 0,1,2,3 (Property of Decimal division).

6. Which of the following x is accepted by the given DFA (x is a binary string ∑= {0,1})? a) divisible by 3
b) divisible by 2
c) divisible by 2 and 3
d) divisible by 3 and 2

Explanation: The given DFA accepts all the binary strings such that they are divisible by 3 and 2.Thus, it can be said that it also accepts all the strings which is divisible by 6.

7. Given:
L1= {xϵ ∑*|x contains even no’s of 0’s}
L2= {xϵ ∑*|x contains odd no’s of 1’s}
No of final states in Language L1 U L2?
a) 1
b) 2
c) 3
d) 4

8. The maximum number of transition which can be performed over a state in a DFA?
∑= {a, b, c}
a) 1
b) 2
c) 3
d) 4

Explanation: The maximum number of transitions which a DFA allows for a language is the number of elements the transitions constitute.

9. The maximum sum of in degree and out degree over a state in a DFA can be determined as:
∑= {a, b, c, d}
a) 4+4
b) 4+16
c) 4+0
d) depends on the Language

Explanation: The out degree for a DFA I fixed while the in degree depends on the number of states in the DFA and that cannot be determined without the dependence over the Language.

10. The sum of minimum and maximum number of final states for a DFA n states is equal to:
a) n+1
b) n
c) n-1
d) n+2

Explanation: The maximum number of final states for a DFA can be total number of states itself and minimum would always be 1, as no DFA exits without a final state. Therefore, the solution is n+1.

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