Translate each of these quantifications into english and determine its truth value.

a.∃x∈r (x3 = - 1)

b.∃x∈z (x + 1 > x)

c.∀x∈z (x - 1 ∈ z)

d.∀x∈z (x2 ∈ z)

a) ∃x∈r (x3 = - 1)

English: There exists an x belongs to Real Space, such that x3 = - 1

Test for truth value: we have x = - 1 which belongs to R

Such that x3 = (-1) 3 = - 1. - True

b) ∃x∈z (x + 1 > x)

English: There exists an x belongs to Integers, such that x+1 > x

Test for truth value: we have x=1 which is an integer

Such that x+1 = 1+1 = 2 > 1. - True.

c) ∀x∈z (x - 1 ∈ z)

English: For all x belonging to integers, x-1 also belongs to integer.

Proof:

Define a function f (x) = x-1

Domain of function is Z

The range of functions is also Z. because there exists a one-to-one mapping.

Hence True

d) ∀x∈z (x2 ∈ z)

English: For all x belonging to integers, x2 also belongs to integer.

Proof:

Define a function f (x) = x2

Now Z is - infinity ...-2,-1,0,1,2, ... infinity

Range of the function will be 0,1,4,9,16,25,36 ... infinity, which are all integers

So, True