Question Video: Operations on Number Sets Mathematics • 8th Grade

What is ℚ ∩ ℚ′?

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Video Transcript

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 it cannot.

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.