Question

a student made the following false statement: \all triangles are isosceles.\ which of the statements below is the negation of the above statement? a. all triangles are not isosceles. b. not all triangles are isosceles. c. all triangles are scalene or equilateral.

Explanation:

Step1: Understand the negation of a universal statement

The original statement is “All triangles are isosceles”. The negation of a universal statement “All \(x\) are \(P(x)\)” is “There exists an \(x\) such that not \(P(x)\)”. In other - words, it is “Not all triangles are isosceles”.

Step2: Analyze each option

• Option A: “All triangles are not isosceles” is a different (and incorrect) form of negation. It implies a universal non - property, which is not the correct negation of the original statement.
• Option B: “Not all triangles are isosceles” is the correct negation. It means that there is at least one triangle that is not isosceles.
• Option C: “All triangles are scalene or equilateral” is not the negation of the original statement. The original statement is about the property of being isosceles, and this option is about classifying triangles into scalene or equilateral, which is a different classification (though scalene and equilateral are non - isosceles in a sense, the form of the statement is not the logical negation).

Answer:

B. Not all triangles are isosceles.