Video Transcript
What is ℚ minus ℚ prime?
To answer this question, we might
be tempted to start by recalling that ℚ is the set of rational numbers, that is, the
set of all quotients of integers where the denominator is nonzero. However, it is not actually
necessary to do this to answer this question. We can also note that the prime
symbol in this context means the complement of the set, that is, all of the numbers
that are not rational. This may also be written with a
superscript 𝑐.
The easiest way to answer this
question is to recall that for any set 𝐴, the set 𝐴 and its complement share no
elements. So subtracting the complement of 𝐴
from 𝐴 will leave 𝐴 unchanged. Applying this result with 𝐴 equal
to the set of rational numbers gives us that the set of rational numbers minus the
set of irrational numbers is equal to the set of rational numbers.
While this is the easiest way to
answer this question, it can be useful to actually think about this in terms of the
definitions of the sets. Subtracting the set of irrational
numbers from the set of rational numbers means that we need to remove every
irrational number from the set of rational numbers. Of course, there are no irrational
numbers in the set of rational numbers. So once again, we can say that
subtracting the set of irrational numbers from the set of rational numbers leaves us
with the set of rational numbers.
