Question

Select the logical expression that is equivalent to ¬∃x(p(x)∧q(x))?

1) ∀x(¬p(x)∨¬q(x))
2) ∀x(¬p(x)∧¬q(x))
3) ∀x(p(x)∨q(x))
4) ∀x(p(x)∧q(x))

Answer 1

The equivalent logical expression to ¬∃x(p(x)∧q(x)) is ∀x(¬p(x)∨¬q(x)), according to De Morgan's laws.

The correct answer is 1).

Step-by-step explanation:

The student is asking for the equivalent logical expression to ¬∃x(p(x)∧q(x)), which in plain language means 'there is no x for which both p(x) and q(x) are true'. To find an equivalent expression, we can use De Morgan's laws. De Morgan's laws state that the negation of an existential quantifier combined with a conjunction is equivalent to a universal quantifier combined with a disjunction of the negations. This gives us the expression ∀x(¬p(x)∨¬q(x)), which translates back to 'for all x, either p(x) is not true or q(x) is not true'.

Therefore, the equivalent logical expression is Option 1: ∀x(¬p(x)∨¬q(x)).

Answer 2

The logical expression that is equivalent to ¬∃x(p(x)∧q(x)) is ∀x(¬p(x)∨¬q(x)).

Step-by-step explanation:

The logical expression that is equivalent to ¬∃x(p(x)∧q(x)) is 1) ∀x(¬p(x)∨¬q(x)).

To understand why this is the correct answer, let's break it down:
¬∃x(p(x)∧q(x)) translates to "It is not true that there exists an x such that p(x)∧q(x)". Using De Morgan's law, we can rewrite this as ∀x(¬p(x)∨¬q(x)), which means "For all x, it is not true that p(x)∧q(x)".

The other options do not accurately represent the negation of ∃x(p(x)∧q(x)) and are therefore not equivalent to it.