JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Mathematical Reasoning *

Which of the following is the negation of the statement "for all

M > 0

, there exists

x \in S

such that

x \geq M

Answer: (D)

Solution

The statement is

\forall M > 0, \exists x \in S

such that

x \geq M

. Negation turns a universal quantifier into an existential one and an existential into a universal, and replaces

x \geq M

x < M

. The negation is therefore

\exists M > 0

such that

x < M

for every

x \in S

, that is: there exists

M > 0

such that

x < M

for all

x \in S

JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Single correct option