વિધેય \(f : N \to N\) ; \(f\left( x \right) = x - 5\left[ {\frac{x}{5}} \right]\) ,કે જ્યાં \(N\) એ પ્રાકૃતિક સંખ્યા નો ગણ છે અને \([x]\) એ મહતમ પૃણાંક વિધેય છે તો વિધેય . .. .

A એક-એક અને વ્યાપ્ત છે .
B એક-એક છે અને વ્યાપ્ત નથી.
C એક-એક નથી અને વ્યાપ્ત છે .
D એક-એક અને વ્યાપ્ત બને નથી.

Verified Solution

Answer & Solution

Correct Answer

(D) એક-એક અને વ્યાપ્ત બને નથી.

Step-by-step Solution

Detailed explanation

\(\left. \begin{array}{l} f\left( 1 \right) = 1 - 5\left[ {1/5} \right] = 1\\ f\left( 6 \right) = 6 - 5\left[ {6/5} \right] = 1 \end{array} \right\} \to \) Many one \(f\left( {10} \right) = 10 - 5\left( 2 \right) = 0\) which is not in co-domain. Neither one-one nor onto.

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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