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

c is the answer

step-by-step explanation:

in mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. there are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. the set of candidate solutions that satisfy all constraints is called the feasible set.

hoped i'm right lol