Negate the following expression and select the best answer below: A = for all; E = for some.*

To negate the given expression using logical quantifiers, let's break down the problem as follows:

1. Break Down the Problem

• The original expression involves two quantifiers:
  • "A" represents "for all" (universal quantifier).
  • "E" represents "for some" (existential quantifier).
• We need to determine the negation of a statement involving these quantifiers.

2. Relevant Concepts

• The negation of a universal quantifier can be expressed as an existential quantifier:
  • $\neg (\forall x, P(x)) \equiv \exists x, \neg P(x)$
• The negation of an existential quantifier can be expressed as a universal quantifier:
  • $\neg (\exists x, P(x)) \equiv \forall x, \neg P(x)$

3. Analysis and Detail

• If we assume the expression is something like $\forall x, P(x)$ (for all x, P holds), its negation would be:

  • $\exists x, \neg P(x)$ (there exists at least one x for which P does not hold).

• If the expression is something like $\exists x, P(x)$ (there exists at least one x such that P holds), its negation would be:

  • $\forall x, \neg P(x)$ (for all x, P does not hold).

4. Verify and Summarize

• To summarize:
  • Negating "for all" results in "for some not."
  • Negating "for some" results in "for all not."

Final Answer

The negation of an expression using these quantifiers can generally be expressed as follows:

• Negation of $\forall x, P(x)$ is $\exists x, \neg P(x)$
• Negation of $\exists x, P(x)$ is $\forall x, \neg P(x)$

Select the appropriate negated expression based on the context provided.