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ava789
10.07.2019 •
Mathematics
Prove that the set of even integers is denumerable
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Ответ:
to show that the set of all even integers is denumerable, we prove that there is a bijection from
to
, where
, the set of all even integers.
proof. note that the set of even integers can be put into a list, that is,
. we can intuitively see that all the even integers can be listed, allowing for a way to count all the even integers (however, we will never finish counting). we can be more rigorous in our argument by finding an explicit bijection. we can think about how to recreate the list we have made.
let the function
be defined by
for example,
,
,
,
, and so on. clearly this function generates this list. now we show that
is a bijection. to do this, we must show that
is injective and surjective.
injectivityto show that
is injective, we must show that for any
,
implies
.
suppose that
such that
. then we have four cases:
case 1: if
are both odd, then
case 2: if
are both even, then
case 3: if
is even and
is odd, then
but since
is a positive even integer, we have
and since
is a positive odd integer, we have
.
we inferred that
is true in this case, but the left- and right-hand inequalities actually do not agree. this is a contradiction, so case 3 cannot happen.
case 4: if
is odd and
is even, then by similar argument as in case 3, case 4 cannot happen.
hence
is injective.
surjectivitynow we show that
is a surjection. to do this, we must show that for any
, there is
such that
. we must assert that there is an element in the domain that has
as its image. so suppose we have
. here,
is even.
case 1: if
, then
and
, since
is even.
case 2: if
, then
so
and
, since
is odd.
case 3: if
then
.
so
is also a surjection.
therefore,
is a bijection because it is injective and surjective.
it follows that
, the set of all even integers, is denumerable.
Ответ:
N = 180
Step-by-step explanation:
Of means multiply and is means equals
25% * N = 45
Change to decimal form
.25 N = 45
Divide each side by .25
.25N/.25 = 45/.25
N = 180