What does the given DFA represent in the example $\Sigma = \{ a,b, c\} $ with states $Q= \{q,r,s,t \}$?

Question

BlackTom AI · Accepted Answer

The question asks: What does the given DFA represent in the example Σ = { a, b, c } with states Q = { q, r, s, t }?

Since the provided answer option is 'c. Language with the word 'abc'', we will evaluate the idea behind this choice and also note the absence of other options.

First, consider the general task: a DFA over the alphabet {a, b, c} recognizes a language—i.e., a set of strings over that alphabet. The specific DFA in the example could represent several possible languages depending on its start state, accepting states, and transition function. If the intended interpretation is that the DFA accepts exactly the string 'abc', that would be a valid language recognized by a DFA (the singleton language {abc}). In such a case, the automaton would start in the start state, move through transitions on a -> b -> c, and end in an accepting state.

Now, examining option 1 (which is the only option provided): 'c. Language with the word 'abc'' suggests the DFA recognizes all strings that contain the exact substring or exact word 'abc' as the entire string or as a portion depending on design. If the DFA is designed so that only the string 'abc' leads to an accepting state and all other strings lead to a non-accepting state, then the language recognized is exactly {abc}, which is commonly referred to as the singleton language consisting of the word abc.

If there were other options, we would compare them by considering typical DFA interpretations in such exercises: (1) a language of all strings, which would require that every path ends in an accepting state regardless of input; (2) a language consisting of all strings ending with abc, which would require loops and specific final transitions; (3) an empty language, which would mean no accepting paths exist; (4) a language containing any of several specific words, etc. In those cases, the structure of the DFA—start state, accepting states, and transitions—clearly determines the recognized language.

Because only one option is present here and it asserts that the DFA represents the language containing the word abc, this aligns with the common interpretation that the DFA recognizes a very specific string when designed accordingly. In summary, the provided option identifies a plausible and common interpretation for such a DFA over {a, b, c} with four states.

Note: If additional options existed, each would be evaluated against the actual transition structure to determine whether it matches the described language. The absence of other options limits the comparative analysis to the given choice and general DFA-language correspondence.

CSC 333 (002) SPRG 2025 MCQ2

What does the given DFA represent in the example \(\Sigma = \{ a,b, c\} \) with states \(Q= \{q,r,s,t \}\)?

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