Question

g . 3. For each of the following statements Write the statement as an English sentence that does not use the symbols for quan 2.4. Quantifiers and Negations 75 Write the negation of the statement in symbolic form in which the negation symbol is not used. Write a useful negation of the statement in an English sentence that does not use the symbols for quantifiers.

Answer 1

(a)

( ∃x ∈ Q) ( x > √2)

There exists a rational number x such that x > √2.

( ∀x ∈ Q) ( ( x ≤ √2)

For each rational number x, x ≤ √2.

(b)

(∀x ∈ Q)(x² - 2 ≠ 0).

For all rational numbers x, x² - 2 ≠ 0

( ∃x ∈ Q ) ( x² - 2 = 0 )

There exists a rational number x such that x² - 2 = 0

(c)

(∀x ∈ Z)(x is even or x is odd).

For each integer x, x is even or x is odd.

( ∃x ∈ Z ) (x is odd and x is even)

There exists an integer x such that x is odd and x is even.

(d)

( ∃x ∈ Q) ( √2 < x < √3 )

There exists a rational number x such that √2 < x < √3

(∀x ∈ Q) ( x ≤ √2 or x ≥ √3 )

For all rational numbers x, x ≤ √2 or x ≥ √3.