Question

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.

Answer 1

The rewritten statements is as follows:

(a) ∀ dinosaur x, x is extinct.

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

(c) ∀ irrational
`number` x, x is not an
`integer`.

(d) ∀ logician x, x is not lazy.

(e) ∀ integer x,
`x^2 \\e 2,147,581,953`

(f) ∀ real number x,
`x^2 \\e -1`

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)

`All \to` ∀

`dinosaurs \to x`

(b)

`Every \to` ∀

`real\ number \to x`

(c)

First, we need to rephrase this sentence to:

Every irrational number is not integer

So, the replacement is:

`Every \to` ∀

`irrational\ number \to x`

(d)

First, we need to rephrase this sentence to:

Every logician is lazy

So, the replacement is:

`Every \to` ∀

`lazy \to x`

(e)

First, we need to rephrase this sentence to:

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

So, the replacement is:

`The \to` ∀

`Integer \to x`

(f)

Replace:

`The \to` ∀

`-1 \to x^2`