Quantifiers?

Question

For each of the following statements
a-write as an englsh sentence that does not use the symbols for quantifiers
b-Write the negation of the statement in symbolic form in which the negation symbol is not used.
c-Write a useful negation of the statement in an English sentence that does not use the symbols for quantifiers.
d-Also tell if TRUE or FALSE

√= square root

1-(∀x∈ Z) ( x is even or x is odd)
2-(∃x ∈ Q) ( √2 3-(∀x ∈ Z) ( if x2 is odd, then x is odd)
4-(∀n∈N) (n2-n+41 is a prime number)

Answer 1

1.
a) Every integer is even or odd.
b) (∃x∈ Z) (x is neither even nor odd)
c) There is an integer that is neither even nor odd.
d) The original statement is true; the negation is false.

2.
a) There is a rational number that is strictly between √2 and √3.
b) (∀x ∈ Q) (√2 > x or x > √3)
c) All rational numbers are less than √2 or greater than √3.
d) The original statement is true; the negation is false.

3.
a) Every integer whose square is odd is also itself odd.
b) (∃x ∈ Z) ( x² is odd and x is even)
c) There is an even integer whose square is odd.
d) The original statement is true; the negation is false.

4.
a) The result of squaring any counting number, subtracting itself, and adding 41 is always a prime number.
b) (∃n∈N) (n²-n+41 is a composite number)
c) There is a counting number that, after being squared, having itself subtracted, and adding 41, results in a composite number.
d) The original statement is false; its negation is true. (4 is such a number.)

Answer 2

1. a) All integers are either even or odd.
b) (∃x ∈Z) ( x is not even and x is not odd)
c) There is an integer which in neither even nor odd.
d) True

2. a) There is a rational number between √2 and √3
b) (∀x∈Q) ( x<√2 or x>√3)
c) There exist no rational numbers between √2 and √3
d) True

3. a) For all integers, x^2 is odd implies x is odd
b) (∃x ∈ Z) (x^2 is odd and x is even)
c) There exists an even integer such that its square is odd.
d) True

4. a) For all natural numbers n, n^2-n+41 is a prime number.
b) (∃n ∈ N) (n^2-n+41 is composite)
c) There exists a natural number n such that n^2-n+41 is
composite number.
d) False. If n = 41, that 41^2 - 41+41 = 41*41, so it is not prime.

Answer 3

1-
a)for all x in Z, x is either even or odd
b)(∃x ∈ Z) (x is even and x is odd)
c)there is an x in Z such that x is even and odd
d)TRUE
2-
a)There exists an x in Q such that x is greater than √2 and less than √3
b)(∀x∈Q)(x<=√2 or x>=√3)
c)for all x in Q, x is either less than √2 or greater than √3
d)TRUE (if Q has the regular meaning of rational numbers)
3-
a)for all x in Z, if x^2 is odd, then so is x
b)(∃x∈Z)(x^2 is odd and x is even)
c)There is an x in Z such that x^2 is odd although x is even
d)TRUE
4-
a)For all n in N, n2-n+41 is a prime number
b)(∃x∈N)( n2-n+41 is a composite number)
c)There is an x in N such that n2-n+41 is a composite number
d) FALSE