Video Transcript
What is the set of real numbers
union the set of rational numbers?
In this question, we are given an
expression of the union of two sets and asked to evaluate this expression.
To answer this question, we can
start by recalling what is meant by each of the two sets in the expression. First, we recall that the set of
rational numbers is the set of all quotients of integers where the denominator is
nonzero. Next, we can recall that the set of
real numbers is the union between the set of rational numbers and its complement,
the set of irrational numbers.
This then gives us two different
ways to answer this question. First, we can substitute this
expression for the set of real numbers into the given expression to obtain the set
of rational numbers union the set of irrational numbers union the set of rational
numbers. Next, we can recall that the union
of sets is an associative and commutative operation. So we can reorder this union to get
the union of the set of rational numbers with itself union the set of irrational
numbers.
We know that taking a union between
a set and itself gives that set. So the expression simplifies to
give us the union of the set of rational and irrational numbers. This is the definition of the set
of real numbers.
There is a second method we can use
to answer this question. We can note that the set of
rational numbers is a subset of the set of real numbers. We can then recall that taking a
union between a set 𝐵 and a subset of this set 𝐴 will give 𝐵. Therefore, since the set of
rational numbers is a subset of the set of real numbers, taking the union of these
sets will still give us the set of real numbers.
