elementary set theory – Proof of |S U T| = |S| + |T| – |S ∩ T|

I’m asked to prove the statement in the title under the assumption that I do not know the Inclusion-Exclusion Principle. I have two ways of starting the proof where:

  1. I could declare two sets with a certain amount of values and show by example that it is true:

A = {1. 2, 3, 4} and B = {3, 4, 5, 6}

|A| = 4, |B| = 4

|A ∪ B| = |A| + |B| – |A ∩ B| = 4 + 4 – 2 = 6

  1. I could state that it is true and give a logical explanation:

This is true, because to count the number of elements in A ∪ B, we start by counting those in A, and then add those in B. If A and B were disjoint, then we are done, otherwise, we have double counted those in both sets, so we must subtract those in A ∩ B.

However, I don’t know if these are counted as formal proofs. If not, how would I start a proof like this?

elementary number theory – Is it possible to improve the resulting upper bound for $dfrac{D(m)}{s(m)}$, given a lower bound for $I(m)$?

Let $sigma=sigma_{1}$ be the classical sum of divisors. For example,
$$sigma(12)=1+2+3+4+6+12=28.$$

Define the following arithmetic functions:
$$D(n)=2n-sigma(n)$$
$$s(n)=sigma(n)-n$$
$$I(n)=dfrac{sigma(n)}{n}.$$

Here is my initial question:

QUESTION

Is it possible to improve the resulting upper bound for $dfrac{D(m)}{s(m)}$, given a lower bound for $I(m)$?

MY ATTEMPT

For example, assume that a lower bound for $I(m)$ is given as
$$I(m) > c$$
where $1 < c in mathbb{R}$.

We rewrite
$$dfrac{D(m)}{s(m)}$$
as
$$dfrac{D(m)}{s(m)}=dfrac{2m-sigma(m)}{sigma(m)-m}=dfrac{2-I(m)}{I(m)-1} < frac{2-c}{c-1},$$
since
$$bigg(I(m) – 1 > c – 1bigg) land bigg(2 – I(m) < 2 – cbigg) iff dfrac{2-I(m)}{I(m)-1} < frac{2-c}{c-1}.$$

Here is my follow-up question:

Can we do better than the upper bound
$$dfrac{D(m)}{s(m)} < frac{2-c}{c-1},$$
if $I(m) > c$ (where $1 < c in mathbb{R}$)?

I am under the impression that one can come up with a tighter bound.

calculus and analysis – It is possible to express the error function in term of others elementary functions?

Please, some help.
I obtained using Mathematica some results written in terms of the error function. Erfi(x)
The question is there is a way to transform the error function in other’s special functions i.e Bessel, or others. I wonder if it can be done using Mathematica. Any suggestion is welcome

U= Erfi(((1/2 + I/2) (R – z))/Sqrt(k R)) + Erfi(((1/2 + I/2) (R – Sqrt(D^2 + z^2)))/Sqrt(k R))

Meaning of “Dependency” in Relation to Elementary DB Normalization

In trying to figure out why the following table definitions from a textbook exercise about nurses in hospital wards is not in 2NF:

Ward (WName, Location, WType)

Ward-Nurse (WName(fk), NurseID, NurseName, TeamCode, TeamSkill,
Shift)

what is the meaning of “depends on” when describing how some attributes don’t “depend” on the whole key? Is it something like “you don’t need to know the ward name in order to know what shift a nurse is working” for example?

What is the thought process by which it becomes obvious that for this DB to be in 2NF I need to change the definition to

Ward ( WName, Location, WType)

Ward-Nurse ( WName(fk), NurseID(fk), Shift)

Nurse ( NurseID, NurseName, TeamCode, TeamSkill)

Likewise for 3NF, what interpretation of the word “depends” and its application to the table definitions above allows me to know that the solution for 3NF is

Ward (WName, Location, WType)

Ward-Nurse ( WName(fk), NurseID(fk), Shift)

Nurse ( NurseID, NurseName, TeamCode(fk))

Team (TeamCode, TeamSkill)

All the above being from a textbook exercise where no explanation is given beyond the proposed table definitions.

elementary set theory – What ordinal numbers have to do with 1st, 2nd, 3rd etc?

I Googled “ordinal numbers” and stuff for kids came up, on a website I found

“An Ordinal Number is a number that tells the position of something in a list, such as 1st, 2nd, 3rd, 4th, 5th etc”

So my question is, what do ordinal numbers in this sense have to do with ordinal numbers in Cantor’s set theory?

elementary number theory – Prove that if $dmid c$ then $gcd(a,b)=gcd(a+c,b)$.

My solution is as follows:

Let $d_1= (a,b) > 0$. Then by Bezout’s identity, there exists integers $x_1$and $y_1$ such that, $d_1=ax_1+by_1$. Similarly let $d_2 = (a+c,b) > 0$ then there exists integers $x_2$ and $y_2$ such that,
$d_2=(a+c)x_2+by_2$. Then since $bmid c$ there exists integer $k$ such that $c=bk$ for some $k$.

Note that, $d_2=(a+c)x_2+by_2$. Since $c = bk$ then, $d_2=(a+bk)x_2+by_2$ $Rightarrow$ $d_2 = ax_2+b(kx_2+y_2)$, which is a linear combination in $a$ and $b$. Since $d_1 = (a,b)$, it divides any linear combination in $a$ and $b$. Thus $d_1mid d_2$.

Similarly, $d_1=ax_1+by_1$ $Rightarrow$ $d_1=ax_1+by_1+cx_1-cx_1$. Since, $c=bk$ $Rightarrow$ $d_1=(a+c)x_1+b(y_1-kx_1)$. Which is a linear combination in $a+c$ and $b$. Thus $d_2mid d_1$.

Since $d_1$ and $d_2$ are $gcd$ of two integers, and $d_1mid d_2$ and $d_2mid d_1$, we conclude that $d_1=d_2$ as desired.

Thank you for reading upto here. If there is any errors feel free to comment.

Conitional probability elementary problems – Mathematics Stack Exchange

I have some elementary problem understanging conditional probabliity and expectation with reference to different measure spaces, and maybe the whole theory.

Let’s take this problem as an example for attention:

$X,Y$ are iid exponentially distributed random variables with parameter $lambda$ on probability space $(Omega, mathrm F, mathrm P)$.
Let $Z=min{X,Y}$. Compute $mathrm E(Z|X+Y=M)$ for given M.

It is straightforward to compute PDF $g(x, y)$ of $(X,Y)$, since they are independent, and it’s just $g_x cdot g_y$ if $g_x, g_y$ are $X, Y$ densities respectively.
Rewriting $ Z=f(X,Y)=min{X, Y}$, given that, expectation of $Z|X+Y=M$ would be

$$mathrm E(Z|X+Y=M)=mathrm E(f(X, Y)|X+Y=M)=frac{int_{{(x,y): x+y=M}}{f(x,y) cdot g(x,y) mathrm d(x,y)}}{ mathrm P(X+Y=M)}$$

and
$$ mathrm P(X+Y=M) = int_{{omega: X(omega)+Y(omega)=M}}{mathrm dP_{(X,Y)}} $$

where $ mathrm dP_{(X,Y)} $ is measure over the space on which $(X,Y)$ is defined.

But now, integral $int_{{(x,y): x+y=M}}{f(x,y) cdot g(x,y) mathrm d(x,y)}$ should be equals to zero, because we are integrating over line which has zero Lebesgue measure, and the same with the denominator.

Could you please explain me or give some pdf with theory where i’m making mistake while switching to different measure space? eg. from $P$ to (implicitly) lebesgue? Or link some of these ideas above to Analysis Course which i think i understand more intuitively (at least when it comes to switching from integrals on manifolds to basic integrals)

co.combinatorics – Computing the number of elementary abelian p-subgroups of rank 2 in $GL_{n}(mathbb{F}_{p})$

Let $p$ be a prime number, and let $mathbb{F}_{p}$ be a finite
field of order $p$. Let $GL_{n}(mathbb{F}_{p})$ denote the general linear
group and $U_{n}$ denote the unitriangular group of $ntimes n$ upper
triangular matrices with ones on the diagonal, over the finite field $%
mathbb{F}_{p}$
. In fact $U_{n}$ is a Sylow $p$-subgroup of $GL_{n}(mathbb{F}_{p})$ of order $p^{frac{n(n-1)}{2}}$. Given $n_p^{2}$ be the number of elementary abelian p-subgroups of rank $2$ in $U_{n}$. How can we deduce the number of elementary abelian p-subgroups of rank $2$ in the whole linear group $GL_{n}(mathbb{F}_{p})$?.

Conversely, given $N_p^{2}$ be the number of elementary abelian p-subgroups of rank $2$ in $GL_{n}(mathbb{F}_{p})$. Is there a criterion deduces the number of elementary abelian p-subgroups of rank $2$ in $U_{n}$?.

In other words, what is the relationship between $N_p^{2}$ and $n_p^{2}$?.

Any help would be appreciated so much. Thank you all.

elementary set theory – Is there a concept already for this example which is similar but not homomorphism?

I spot something from an example, which might be a concept already defined, and similar to but not homomorphism.

Roughly, given two relations $R_s subseteq S_1 times S_2$ and $R_t subseteq T_1 times T_2$, for a mapping $f_1: S_1 to T_1$ , there exists $f_2: T_2 to S_2$, s.t. $$ s_1 R_s s_2 text{ iff }t_1 R_t t_2,$$ $forall s_1 in S_1, s_2 in S_2, t_1 in T_1, t_2 in T_2$.

The example is the theorem of syntactic interpretation (Theorem VIII.2.2 in Ebbinghaus’ Mathematical Logic).

I was wondering if there is a concept (in set theory, category theory, …) already for the relationship between $f_1$ and $f_2$, with respect to relations $R_s$ and $R_t$?

Thanks.