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
<span>So,
True</span>
