Question

The negative of "if a student pass in mains then he can write the advance paper" is
1) If a student pass in mains then he can’t write advance papers.
2) If a student fails in mains then he can write the advance paper.
3) A student fails in mains then he can’t write the advance paper.
4) Student pass in mains then he can’t write the advance paper.

Answer 1

Hint: Convert the given statement to a mathematical Boolean equation. Use the rules of negation to solve the equation. Interpret the equation after negating to find the required solution.

Complete step-by-step answer:

The given statement is that "if a student pass in mains then he can write the advance paper" can be converted to a mathematical Boolean equation by assigning variable to the discrete statements.
Let us consider the statement "a student pass in mains" be represented by a Boolean variable\[A\], and the statement "he can write the advance paper" be represented by another Boolean variable\[B\].

Thus the given statement "If A, then B" can be written mathematically as,
\[A \to B\].
Also, it is known that $p \to q$ can be written as $\neg p \vee q$.
Therefore the given statement becomes
$\neg A \vee B$
On negating the statement, we get
$\neg \left( {\neg A \vee B} \right)$
We can simplify the expression using De Morgan's law
$A \wedge \neg B$
Converting it to the statement, we get
Students pass in mains and can't write the advance paper.

It can also be rephrased as "A student pass in mains then he can't write the advance paper."
Therefore the option D is correct.

Note: The De Morgan's Law states the $\neg \left( {a \vee b} \right)$is $\neg a \wedge \neg b$. The Boolean statement $p \to q$ can be written as $\neg p \vee q$.