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

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step-by-step explanation: