Question

Let A(x) and B(x) be abstract predicates. Which of the propositions below are equivalent to the proposition ¬(∀x(A(x)⟹B(x)) ?

Select all that apply.
A. ∀x¬(A(x)⟹B(x))
B. ∃x¬(A(x)⟹B(x))
C. ∀x(¬A(x)⟹¬B(x))
D. ∃x(¬A(x)⟹¬B(x))
E. ∀x(¬A(x)∧B(x))
F. ∃x(¬A(x)∧B(x))
G. ∀x(A(x)∧¬B(x))
H. ∃x(A(x)∧¬B(x))

Answer 1

Options A, B, and D are equivalent to the given proposition.

Step-by-step explanation:

Propositional equivalence is a fundamental concept in mathematical logic and is essential in various areas, including proof theory, model theory, and the design of logical systems. Understanding when two propositions are equivalent is crucial for making valid logical deductions and simplifying logical expressions.

The proposition ¬(∀x(A(x)⟹B(x)) is equivalent to the proposition ∀x¬(A(x)⟹B(x)), ∃x¬(A(x)⟹B(x)), and ∃x(¬A(x)⟹¬B(x)).

In other words, options A, B, and D are equivalent to the given proposition.