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.

See below

Step-by-step 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)

answer:

total cost for tiles and paints is $924.  

step-by-step explanation:

we have been given that a community hall is in the shape of a cuboid. the hall is 40m long 15m high and 3m wide.

the paint will be required for 4 walls and ceiling.

let us find area of walls and ceiling.

therefore, the area of walls and ceiling is 1410 square meters.

given: cost for 10 litre of paint is $10 and 10 litre paint covers 25 square meter. therefore,

therefore, the total painting cost is $564.

tiles will be required for floor. let us find the area of floor.

given: 1m squared floor tiles costs $3. so,

therefore, the total tiles cost is $360.

now let us find combined total cost of tiles and paint.

therefore, the combined total cost of tiles and paint is $924.