Question Video: Operations on Number Sets | Nagwa Question Video: Operations on Number Sets | Nagwa

Question Video: Operations on Number Sets Mathematics • Second Year of Preparatory School

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What is ℚ − ℚ′?

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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.

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