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Suppose Z denotes the set of all integers, Z denotes the set of all positive integers, and Z− denotes the set of all negative integers. Similarly R denotes the set of all real numbers, R denotes the set of all positive real numbers, and R− denotes the set of all negative real numbers. Suppose N denotes the set of all natural numbers and Q denotes the set of all rational numbers. Enter "T" for each true, and "F" for each false statements.(a) Z / Q = Z(b) Z / Z- = N

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Answer:

a) False

b) False

Step-by-step explanation:

We are given the following information:

Z is a set of all integers, [tex]Z^+[/tex] is a set of all positive integers and  [tex]Z^-[/tex] is a set of all negative integers.

Q is a set of all rational numbers, [tex]Q^+[/tex] is a set of all positive rational numbers and [tex]Q^-[/tex] is a set of all negative rational numbers.

N is a set of all natural numbers.

a) False

We will show this with the help of a counter example.

[tex]Z = 3\\Q = \frac{3}{4}\\\displaystyle\frac{Z}{Q} = \displaystyle\frac{3}{\frac{3}{4}} = \displaystyle\frac{8}{3}[/tex]

which is a rational number and not an integer.

b) False

We will show this with the help of a counter example.

[tex]Z = 3\\Z^- = -5\\\displaystyle\frac{Z}{Z^-} = \displaystyle\frac{3}{-5} = -\displaystyle\frac{3}{5}[/tex]

which is a rational number and not a natural number.

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