168 Review Questions and Answers
Question 9-5
One of the following statements is false. Which one?- Some real numbers are rational.
- Some real numbers are irrational.
- All integers are rational.
- Some integers are irrational.
- All irrationals are real.
- All integers are real.
Answer 9-5
The fourth statement is false. No integer is irrational. Every integer is rational because it can
be expressed as a ratio of integers. (Any integer is equal to itself divided by the integer 1!)Question 9-6
Which of the sets N, Z, Q, S, and R are denumerable? Which are not?Answer 9-6
The sets N, Z, and Q are denumerable. That means the elements of each of these sets can be
arranged in an “implied list,” even though the “list” can’t be written out in full because it’s
infinitely long. The sets S and R are not denumerable. Their elements can’t be arranged in any
sort of “implied list.” Even an infinitely long “list” can’t capture them all!Question 9-7
What does it mean for an operation to be closed over the set of real numbers?Answer 9-7
Imagine that we have an operation between two quantities. Let’s call that operation “pound”
and use the symbol #. The “pound” operation is closed over the set of real numbers if and only
if, for any two real numbers x and y, the quantity x # y is also a real number.Question 9-8
Which of the common arithmetic operations are closed over the set of reals? Which are not?Answer 9-8
Addition, multiplication, and subtraction are closed over the set of real numbers. Division is
not, because if we divide a real number by 0, we get an undefined quantity. Exponentiation
(the raising of a real number to a real-number power) is not, because 0^0 is undefined.Question 9-9
What about taking a real root of a real number? Is this operation closed over the set of reals?Answer 9-9
No. Any even-integer root of a negative real number produces an imaginary number, and no
imaginary number is real. The 0th root of any real number is undefined, because that’s the
equivalent of taking the number to the power of 1/0.