Q76 / 150 · MHT CET 2025 (25 Apr Shift 1) · Mathematics

The negation of the statement "The triangle is an equilateral or isosceles triangle and the triangle is not isosceles and it is right angled" is

  • A
    The triangle is not an equilateral or not an isosceles triangle or it is not an isosceles or it is not right angled
  • B
    The triangle is not an equilateral triangle or not isosceles triangle and it is isosceles or it is not right angled
  • Correct answer
    C
    If the triangle is an equilateral triangle or an isosceles triangle then it is an isosceles triangle or not right angled
  • D
    If the triangle is an equilateral triangle or an isosceles triangle then it is not isosceles triangle and it is not right angled

Solution

Define the propositions:
P: the triangle is equilateral
Q: the triangle is isosceles
R: the triangle is right-angled

The statement $S$ is $(P \lor Q) \land (\neg Q \land R)$.

Applying De Morgan's Law to the negation:
$\neg S \equiv \neg[(P \lor Q) \land (\neg Q \land R)]$
$\equiv \neg(P \lor Q) \lor \neg(\neg Q \land R)$

Applying De Morgan's Law to each disjunct:
$\neg(P \lor Q) \equiv \neg P \land \neg Q$
$\neg(\neg Q \land R) \equiv \neg\neg Q \lor \neg R \equiv Q \lor \neg R$

Combining these yields:
$\neg S \equiv (\neg P \land \neg Q) \lor (Q \lor \neg R)$

Option C expresses the implication $(P \lor Q) \to (Q \lor \neg R)$, which is logically equivalent to $\neg(P \lor Q) \lor (Q \lor \neg R)$
Applying De Morgan's Law gives $(\neg P \land \neg Q) \lor (Q \lor \neg R)$

This matches our derived negation exactly, making Option C correct.

Hence, the correct answer is option: 'C'.

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