I'll give you an example from topology that might help - even if you don't know topology, the distinction between the proof styles should be clear.
Proposition: Let
S
be a closed subset of a complete metric space (,)
(
E
,
d
)
. Then the metric space (,)
(
S
,
d
)
is complete.
Proof Outline: Cauchy sequences in (,)
(
S
,
d
)
converge in (,)
(
E
,
d
)
by completeness, and since (,)
(
S
,
d
)
is closed, convergent sequences of points in (,)
(
S
,
d
)
converge in (,)
(
S
,
d
)
, so any Cauchy sequence of points in (,)
(
S
,
d
)
must converge in (,)
(
S
,
d
)
.
Proof: Let ()
(
a
n
)
be a Cauchy sequence in (,)
(
S
,
d
)
. Then each ∈
a
n
∈
E
since ⊆
S
⊆
E
, so we may treat ()
(
a
n
)
as a sequence in (,)
(
E
,
d
)
. By completeness of (,)
(
E
,
d
)
, →
a
n
→
a
for some point ∈
a
∈
E
. Since
S
is closed,
S
contains all of its limit points, implying that any convergent sequence of points of
S
must converge to a point of
S
. This shows that ∈
a
∈
S
, and so we see that →∈
a
n
→
a
∈
S
. As ()
(
a
n
)
was arbitrary, we see that Cauchy sequences in (,)
(
S
,
d
)
converge in (,)
(
S
,
d
)
, which is what we wanted to show.
The main difference here is the level of detail in the proofs. In the outline, we left out most of the details that are intuitively clear, providing the main idea so that a reader could fill in the details for themselves. In the actual proof, we go through the trouble of providing the more subtle details to make the argument more rigorous - ideally, a reader of a more complete proof should not be left wondering about any gaps in logic.
(There is another type of proof called a formal proof, in which everything is derived from first principles using mathematical logic. This type of proof is entirely rigorous but almost always very lengthy, so we typically sacrifice some rigor in favor of clarity.)
As you learn more about a topic, your proofs typically begin to approach proof outlines, since things that may not have seemed obvious before become intuitive and clear. When you are first learning it is best to go through the detailed proof to make sure that you understand everything as well as you think you do, and only once you have mastered a subject do you allow yourself to omit obvious details that should be clear to someone who understands the subject on the same level as you.
The type of data that would be best displayed in a box plot is the number of votes received by each of four candidate.
<h3>What is a box plot?</h3>
A box plot is used to study the distribution and level of a set of scores. The scores are distributed into 4 groups. Each group has a value of 25%
The whiskers represent the minimum and maximum scores. On the box, the first line to the left represents the lower (first) quartile. The next line on the box represents the median. The third line on the box represents the upper (third) quartile.
To learn more about box plots, please check: brainly.com/question/27215146
#SPJ1
Answer:
Step-by-step explanation:
<u>Given</u>
<u>To find</u>
<u>Solution</u>
- P(x) = 0
- cx + c = 0
- cx = -c
- x = -c/c
- x = -1
Hello!
To solve this exercise, we must use the formula below:
Knowing these values, let's replace them in the formula to obtain r:
So, the radius of this circle is 10cm.
Hope this helps!
