Rewrite each of the following statements in the form "∀ x, ." (a) All dinosaurs are extinct. ∀ ---Select--- x, ---Select--- .
(b) Every real number is positive, negative, or zero. ∀ ---Select--- x, ---Select--- .
(c) No irrational numbers are integers. ∀ ---Select--- x, ---Select--- .
(d) No logicians are lazy. ∀ ---Select--- x, ---Select--- .
(e) The number 2,147,581,953 is not equal to the square of any integer. ∀ ---Select--- x, ---Select--- .
(f) The number −1 is not equal to the square of any real number.

The rewritten statements is as follows:

(a) ∀ dinosaur x, x is extinct.

(b) ∀ real x, x is positive, , or zero.

(c) ∀ irrational x, x is not an .

(d) ∀ logician x, x is not lazy.

(e) ∀ integer x,

(f)  ∀ real number x,

Step-by-step explanation:

As a general rule, words like "all", "every", "the", etc. are represented with ∀.

However, some sentences may be need to rephrased before such sentence can be rephrased

So, the following replacement will be made:

(a)

∀

(b)

∀

(c)

First, we need to rephrase this sentence to:

Every irrational number is not integer

So, the replacement is:

∀

(d)

First, we need to rephrase this sentence to:

Every logician is lazy

So, the replacement is:

∀

(e)

First, we need to rephrase this sentence to:

The integer 2,147,581,953...............

So, the replacement is:

∀

(f)  

Replace:

∀

step-by-step explanation:

x^2 -12x + 36 + y^2 = 36

x^2 + y^2 = 12x

r^2 = 12rcos(theta)

r = 12 cos(theta)