What is the intersection between ℚ
and ℚ prime?
In this question, we are asked to
evaluate an expression involving sets.
The easiest way to answer this
question is to recall that the prime symbol means the complement of a set and that
the intersection between any set and its complement is empty. Applying this result with 𝐴 equal
to ℚ gives us that ℚ intersect the complement of ℚ is the empty set. Although this is sufficient to
answer this question, it can actually be useful to consider the meaning of this
result and how we can show it using the definition of ℚ.
We recall that ℚ is the set of
rational numbers, that is, the set of quotients of any two integers such that the
denominator is nonzero. We can then recall that the
complement of this set is called the set of irrational numbers; it contains the
numbers that cannot be written as the quotient of integers.
We can now analyze the intersection
of these sets by considering the properties of any of its elements. Let’s say that 𝑥 is in the
intersection of the set of rational and irrational numbers. For 𝑥 to be in the intersection of
these sets, 𝑥 must be an element of each set. So, 𝑥 must be a rational number
and an irrational number. This is not possible, since 𝑥
either can be written as the quotient of two integers with a nonzero denominator or
Since no element can be in both
sets, we can conclude that their intersection has no elements. In other words, the intersection is
the empty set.