Propositional Logic Language
For each of the following sentences, use the notation introduced in class to convert the sentence into propositional logic. Then write the statement’s negation in propositional logic.
(a) The cube of a negative integer is negative.
(b) There are no integer solutions to the equation x2 −y2 = 10.
(c) There is one and only one real solution to the equation x3 +x+1 = 0.
(d) For any two distinct real numbers, we can find a rational number in between them.
2 Implication
Which of the following implications are always true, regardless of P? Give a counterexample for each false assertion (i.e. come up with a statement P(x,y) that would make the implication false).
(a) ∀x∀yP(x,y) =⇒ ∀y∀xP(x,y).
(b) ∃x∃yP(x,y) =⇒ ∃y∃xP(x,y).
(c) ∀x∃yP(x,y) =⇒ ∃y∀xP(x,y). (d) ∃x∀yP(x,y) =⇒ ∀y∃xP(x,y).
CS 70, Fall 2018, DIS 0B 1
3 Logic
Decide whether each of the following is true or false and justify your answer:
