Exercise · ZW-ORDER-OF-QUANTIFIERS-006 · ★★☆ Intermediate
In general, ∀x∃y R(x,y) and ∃y∀x R(x,y)...
Choose one option.
Show answer
- A. Are not necessarily equivalent
Explanation
The two formulas use the same quantifiers in a different order, and in general they do not have the same truth value. So they are not necessarily equivalent.
Review the concept: Order of quantifiers
For a predicate \(P(x,y)\), the order of the quantifiers can change the truth value of a formula.
\[\begin{gathered}\forall x\,\exists y\,P(x,y)\\\not\equiv\exists y\,\forall x\,P(x,y)\end{gathered}\]
More practice problems: order of quantifiers
- ZW-ORDER-OF-QUANTIFIERS-001 In general, changing the order of ∀ and ∃...
- ZW-ORDER-OF-QUANTIFIERS-002 ∀x ∃y P(x,y) allows...
- ZW-ORDER-OF-QUANTIFIERS-003 ∃y ∀x P(x,y) states that...
- ZW-ORDER-OF-QUANTIFIERS-004 Which pair is not equivalent in general?
- ZW-ORDER-OF-QUANTIFIERS-005 The order of quantifiers matters especially when...
- ZW-ORDER-OF-QUANTIFIERS-007 ∀x∃y R(x,y) allows...
- ZW-ORDER-OF-QUANTIFIERS-008 ∃y∀x R(x,y) states that...
- ZW-ORDER-OF-QUANTIFIERS-009 Select true statements about order.