Answer:
x = 8
Step-by-step explanation:
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 value of the set (A∪C)'∩B is {9 , 12} .
A subfield of mathematical logic called set theory examines sets, which are essentially collections of objects.
- Set theory is a discipline of mathematics that largely addresses problems that are pertinent to mathematics as a whole, despite the fact that any kind of object can be constructed into a set.
- Set theory is founded on a simple binary relationship between an object o and a set A.
- It is written as o A if o is a part (or member) of A. Members of a set are enumerated with commas to denote separation when describing them, or a property of those parts is enclosed in braces.
- Sets can be associated by the membership relation since they are objects.
The given sets are:
U ={1,2,3,4,5,6,7,8,9,10,11,12,13}
A = {3,7,11,13}
B = {3,6,9,12,13}
C = {2,3,5,6,7,8}
Now B' = {1,2,4,5,7,8,10,11}
b) B'∩C = {2,5,7,8,}
c) A∪C = {2,3,5,6,7,8,11,13}
(A∪C)' = {1,4,9,10,12}
(A∪C)'∩B = {9 , 12}
Hence the value of the set (A∪C)'∩B is {9 , 12} .
To learn more about set visit:
brainly.com/question/18877138
#SPJ1
Answer:
The distance between the top of the well and the bottom of the well was 12 1/2
Step-by-step explanation:
(a.k.a B) i did it and got it right :)
