Answer to Question #245879 in Discrete Mathematics for dev

Question

6. Show that (p∧q) → r and (p → r)∧(q → r) are not logically equivalent, without using

truth tables.

7. Express each of these statements into logical expressions using predicates, quantifiers,

and logical connectives. Let the domain consist of all people. Let S(x) be “x is in your

class,” P(x) be “x is perfect.”

(a) Nobody is perfect. (Use only the universal quantifier.)

(b) Nobody is perfect. (Use only the existential quantifier.)

(c) Nobody in your class is perfect.(d) Not everybody is perfect.

(e) Someone in your class is perfect.

(f) Not everyone in your class is not perfect.

Answer 1

6.

Using logical equivalence involving conditional statements:

$(p → r)∧(q → r)\equiv (p\lor q)\implies r$

So, the given statements are not logically equivalent.

7.

a) $\forall x\lnot P(x)$

b) $\nexists x P(x)$

c) $\forall x\lnot P(x)S(x)$

d) $\neg \forall xP(x)$

e) $\exist xP(x)S(x)$

f) $\neg \forall x\neg P(x)S(x)$