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19 June, 21:58

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)

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  1. 19 June, 22:27
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    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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