Question

Given the following sentences, convert them into an appropriate propositional

algebra form, assuming R represents the statement "It is raining", F(x) represents
the predicate "x is a frog", G(x) represents the predicate "x is green", H(x)
represents the predicate "x is hopping", and I(x) represents the predicate "x is
iridescent".
9. Everything is a frog.
10. Nothing is a frog.
11. Green frogs exist.
12. Everything green is a frog.
13. It is raining and some frogs are hopping.
14. If it is raining, then all frogs are hopping.
15. Either everything is a frog or nothing is a frog.
16. Some green frogs are not hopping.

Answer 1

The provided sentences are translated into propositional algebra form using logical symbols representing quantification, conjunction, disjunction, implication, and negation. This interpretation demonstrates propositional and predicate logic to express each statement precisely.

Step-by-step explanation:

Converting the given sentences into propositional algebra form involves representing them using logical predicates, quantifiers, and connectives. Here are the translations for the specified statements:

1. Everything is a frog: ∀x F(x)
2. Nothing is a frog: ∀x ¬F(x)
3. Green frogs exist: ∃x (F(x) ∧ G(x))
4. Everything green is a frog: ∀x (G(x) → F(x))
5. It is raining and some frogs are hopping: R ∧ ∃x (F(x) ∧ H(x))
6. If it is raining, then all frogs are hopping: R → ∀x (F(x) → H(x))
7. Either everything is a frog or nothing is a frog: ∀x F(x) ∨ ∀x ¬F(x)
8. Some green frogs are not hopping: ∃x (F(x) ∧ G(x) ∧ ¬H(x))

Each statement has been converted using the proper logical symbols for universal quantification (∀), existential quantification (∃), logical conjunction (∧), logical disjunction (∨), implication (→), and negation (¬). This representation showcases the propositional and predicative structure of each statement.