Proof Writing – Prove $ X cong R $

To let $ R $ and $ S $ be rings and look at the ring $ R oplus S $, To let $ X = {(x, 0_S): X in R } $, Then, $ X cong R $,

I know a little what I do, but I do not know how to do it.

I know, I have to prove it $ phi: X to R $.
(i) The function is one to one and
(ii) The function is on and
(Iii) $ phi (a + b) = phi (a) + phi (b) $ for all $ a, b in X $, and
(Iv) $ (ab) = ( (a)) ( (b)) $ for all $ a, b in X $,

Can someone please explain to me how I would do that?

Machine Learning – How Do You Prove the Nagarajan Lemma?

To let $ H $ be a hypothesis class of predictors for multiple classes; everyone $ h in H $ is a function of $ X $ to $ (k) $,

Denote the Natarajan dimension of $ H $ by $ Ndim (H) $, Please prove the following lemma.

$ | H | le | X | ^ {Ndim (H)} cdot k ^ {2Ndim (H)} $

The lemma is in the book "Understanding Machine Learning: From Theory to the Algorithm", You are only searching for keywords Lemma 29.4,

Linear Algebra – How do I prove $ R (A) = (Ker (A ^ T)) ^ per $?

To let $ A: mathbb R ^ m to mathbb R ^ n $ be a linear transformation. If $ W $ be a subspace of $ mathbb R ^ n. $ define $$ W ^ perp = {y in mathbb R ^ n | langle x, y rangle = 0 text {for all} x in W } $$
Which of the following statements is correct then?

(on) $ R (A) subset (Ker (A ^ T)) ^ perp $

(B) $ R (A) = (Ker (A ^ T)) ^ perp $

(c) None of the above

Trial: – Claim: – $ R (A) subset (Ker (A ^ T)) ^ perp $

$ z in R (A) implies exists x in mathbb R ^ m: z = A (x) $

We know that $ A ^ T: ( mathbb R ^ n) ^ * to ( mathbb R ^ m) ^ * $, $ W ^ * $ denotes the dual space of $ W. $ Leave, $ Ker (A ^ T) = {g in ( mathbb R ^ n) ^ *: A ^ T (g) = 0 }. $ So, $ (Ker (A ^ T)) ^ perp = {g in ( mathbb R ^ n) ^ * | langle g, h rangle = 0, forall h in Ker (A ^ T) }. $ To let $ g in Ker (A ^ T). $ We have to prove that $ langle g, z rangle = 0 iff langle g, A (x) rangle = 0 $, I can not continue. Please help me. It was a question that had surfaced in the NBHM exam, India. How do I quickly find this complicated answer?

Proof Verification – Suppose $ R $ is a total order for a $ A $ set. Prove that every finite, non-empty set $ B subseteq A $ contains a $ R $ -minest element.

Suggestion:

Accept $ R $ is an overall order on the set $ A $, Prove that every finite,
Non-empty sentence $ B subseteq A $ has a $ R $-minimal element.

My attempt:

Cardinality of the crowd, we say $ A $is described as $ | A | $,

By induction.

Base case:

To take $ B = {b } $ so that $ b in A $, $ b $ is the $ R $-minimal element of $ B $,

Induction step:

Suppose, for all sentences with $ n $ Elements, it will be the smallest element.

Consider any quantity $ B subseteq A $ so that $ | B | = n + 1 $,

Any take $ b in B $,

To let $ B = B setminus {b } $, After the inductive hypothesis $ B & # 39; $ Has $ R $-minimal element. Let's call it $ c $,

Let's look at set $ B $,

Accept $ cRb $, Then $ c $ is the smallest element of $ B $,

Accept $ bRc $, Any take $ y in B $, We know that $ cRy $, Through transitivity we have $ bRy $, $ y $ was therefore arbitrary $ b $ is the smallest element of $ B $,

Suppose that $ lnot bRc $ and $ lnot cRb $, That's impossible because $ R $ is a total order on $ A $,

Therefore not $ c $ or $ b $ becomes the smallest element of $ B $,

$ Box $

Is it right?

formal languages ​​- Prove that $ texttt {Prefix} (L) $ is regular

In the face of that $ L = lbrace 0 ^ n1 ^ n: n geq 0 rbrace $ is an irregular context-free language, prove that $ texttt {prefix} (L) $ is regular.

So far, I have specified the following grammar for creating this language:
$ S rightarrow 0S1 thinspace | thinspace epsilon $

Would you start testing? $ texttt {prefix} (L) $ is just as regular as any other language and proves it $ Sigma ^ star $ = $ texttt {prefix} (L) $or by induction over the length of the words in $ texttt {prefix} (L) $,

Calculus – How do I prove that (x ^ 2) f (x), where f (x) is the Dirichlet function, is no derivative function other than x = 1

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Calculate – Prove that the distance from the manifold to the tangent plane at point x $ o ( varepsilon) is $

Do you have a bit of trouble breaking that,

To let $ M $ be a $ k $-dimensional manifold, $ x in M ​​$ a regular point, and $ v in T_ {x} M $ a vector in the tangential plane.

To prove $ dist (x + varepsilon v, M) = o ( varepsilon) $ as $ varepsilon to $ 0,

I will add that dist refers to the Euclidean distance.

My intuition tells me that $ varepsilon $ approaches zero, the distance from the given point $ x + varepsilon v $ to the distributor must be the distance to the point $ x $But that's obviously wrong, considering what I should prove. Can someone help me find a way to do that?

Agal Algebraic Geometry – How to prove the relation between Pascal's Triangle and the Binomial Theorem

The coefficients of the binomial theorem are based on numbers that appear in the Pascal triangle, where the first term is the degree.
Example:
(a + b) ^ 4 = 1a ^ 4 + 4a ^ 3b + 6a ^ 2b ^ 2 + 4ab ^ 3 + 1b ^ 4,
Where:
1 = 4C0, 4 = 4C1, 6 = 4C2, 4 = 4C3 and 1 = 4C4.
Formula:
Enter image description here

I do not know why the coefficients are probable. Need a proof / explanation of the relationship.
(Incidentally, I'm a beginner, please give me a clear and simple explanation that's easy to understand, thanks!)

Do not Kim Kardashian and Paris Hilton prove without any doubt that capitalism is not meritocracy?

It is not meritocracy at all, but I do not know that they are the proof you are looking for. I'm not a fan of both, but both have worked successfully to build brands independently of their parents. Do not confuse the characters they play in their shows with the women who produce them. Producing a TV show is a torment. Most fiduciary children never do that, they only spend money on their parents until they inherit the rest.

Visa – How can I prove a relationship when my unmarried non-EU partner travels to the UK with a "family member of an EU citizen"?

My partner is Chinese, I'm British, but we both live in the Netherlands. He has a "residence card for a family member of an EU citizen" that we have received, even though we are unmarried because we have lived together for more than 6 months, and who qualified us under EU law.

Next month, we plan to travel to the UK together. As far as I know, he does not need an entry visa when traveling with me. However, the gov.uk website states that it must provide:

"Proof that you are a family member of an EEA citizen (eg your marriage certificate or birth certificate)"

What proof do we have to provide if he has received the residence card referred to in Article 10 without being married?

(I'm just worried we might have borderline issues because he's not a family member, even though he qualifies for the Item 10 card.)