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
In this exercise we have to use the knowledge of mathematical language to write their meanings in English, so we have to:
A)It means that X belongs to the set of reals.
B)It means that X belongs to the set of integers
C)It means that X belongs to the set of integers
D)It means that X belongs to the set of integers
Numerical sets bring together several sets whose elements are numbers. They are formed by the natural, integer, rational, irrational and real numbers. The branch of mathematics that studies numerical sets is set theory.
A) ∃x∈r (x3 = −1) is the same as:
x= -1 which belongs to R
x3 = (-1)3 = -1.
B) ∃x∈z (x + 1 > x) is the same as:
x+1 > x
x=1 which is an integer
x+1 = 1+1 =2 > 1.
C) ∀x∈z (x − 1 ∈ z) is the same as:
x-1 also belongs to integer.
f(x) = x-1
D) ∀x∈z (x2 ∈ z) is the same as:
x2 also belongs to integer.
f(x) = x2
See more about sets at brainly.com/question/8053622
