Whats the expanded form for 79×84?
1 answer:
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Answer:
a) Suppose that F is ordered in ascending order: . Then, the complement of F can be written as
which is the union of a finite number of open intervals, then is an open set. Thus, F is a closed subset of the real numbers.
b) Take an arbitrary element of F, let us say . Now, choose a real number
such that
there are not other element of F, because
is less that the minimum distance between
and its neighbors.
In case that we only consider
, and if
we only consider
.
Then, all points of F are isolated.
Step-by-step explanation:
Two main facts are needed here:
1. The logarithm , regardless of the base of the logarithm, exists for
.
2. The square root exists for
.
(in both cases we're assuming real-valued functions only)
By (2) we know that exists if
, or
.
By (1), we know that exists if
, or
. But as long as the square root exists, it will always be positive, so this condition will always be met.
Ultimately, then, we only require , so the function has domain
.
To determine the range, we need to know that, in their respective domains, and
increase monotonically without bound. We also know that
at minimum, at which point the square root term vanishes, so the least value the function takes on is
. Then its range would be
.