Question

Consider the statement "There are no simple solutions to life's problems."

Write the informal negation of the statement.

(A) At least one of life's problems has a simple solution.
(B) Some of life's problems do not have a simple solution.
(C) All of life's problems do not have a simple solution.
(D) At least one of life's problems does not have a simple solution.
(E) None of life's problems have a simple solution.

Write the formal version of the original statement.

(A) ∀ of life's problems x, x does not have a simple solution.
(B) ∀ of life's problems x, ∃ a simple solution x.
(C) ∃ a simple solution x, ∀ of life's problems.
(D) ∀ of life's problems x, x has a simple solution.
(E) ∃ a simple solution x, for each of life's problems.

Answer 1

See below

Explanation:

Remember the notation and rules of quantifiers. ∀ is the universal quantifier and ∃ is the existential quantifier. To negate ∀x p(x) , write ∃x ¬p(x). To negate ∃x p(x) , write ∀x ¬p(x)

Part I:

A) None of life's problems have a simple solution.

B) All of life's problems have a simple solution.

C) Some of life's problems have a simple solution

D) All of life's problems have a simple solution (notice how the original statements in B and D mean exactly the same)

E) Some of life's problems do not have a simple solution.

Part II: Let x be a variable representing one of life's problems, y be a variable representing solutions, p(x):="x has a simple solution", and q(x,y):="y is a simple solution of x".

A) (∀x)(¬p(x)) or ¬(∃x)(p(x))

B) (∀x)(∃y)(q(x,y))

C) (∃y)(∀x)(q(x,y)). Note that the order of quantifiers is important. B) and C) have different meanings. In C) there is an universal solution of all problems, in B) each problem has its solution.

D) (∀x)(p(x))

E) Same as C)