Analisis Real

We introduce and rigorously prove the Analytic Foundations Theorem (AFT), the first major measure-theoretic generalization of the Fundamental Theorem of Calculus (FTC). The theorem precisely quantifies the discrepancy between integration and differentiation operations in the context of Lebesgue integrable functions, providing explicit bounds on exceptional sets. We extend the results to higher dimensions, probabilistic frameworks, and distributional forms. Applications to differential equations, numerical analysis, and Fuzzy Differentiable Algebraic Graph Systems (F-DAGS) demonstrate the theorem's broad impact.

Dokumen ini merupakan kumpulan tugas dari Kelompok 2 pada mata kuliah Analisis Real yang membahas konsep-konsep dasar dan lanjutan terkait bilangan real, himpunan, fungsi, dan barisan. Setiap anggota kelompok berkontribusi dalam menjelaskan teori, membuktikan teorema, serta menyelesaikan masalah matematis dengan pendekatan yang logis dan sistematis.

Penelitian ini bertujuan untuk melihat  gambaran kemampuan Pengantar Dasar Matematika (PDM), Kalkulus I dan Analisis Real I serta ada atau tidaknya hubungan yang signifikan antara variabel kemampuan PDM dengan Penguasaan Analisis Real I, kemampuan Kalkulus I dengan penguasaan Analisis Reall I dan kemampuan PDM dan Kalkulus I dengan penguasaan Analisis Real I pada mahasiswa prodi pendidikan matematika STAIN Bukittinggi angkatan 2010. Penelitian ini adalah penelitian ex post facto, yaitu penelitian empiris yang sistematis dimana peneliti tidak dapat mengontrol lansung variabel bebas, karena variabel tersebut telah terjadi atau menurut sifatnya tidak dapat dimanipulasi. Hasil penelitian diperoleh kemampuan PDM mahasiswa prodi pendidikan matematika cukup dengan rata-rata 68,75, kemampuan Kalkulus I cukup dengan ratat-rata 65,79 dan penguasaan Analisis Real I sangat kurang (gagal) dengan rata-rata 44,55. Hubungan yang signifikan antara kemampuan PDM dengan penguasaan Analisis real I, kem...

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.

The study aimed at describing university students' ability in constructing mathematical teaching An open mathematical logic and construction of mathematical proof administered t at Mountains of the Moon university analyzed measured for develope mathematics group was knowledge of mathematical logic and method of proof was from to be 2.2 (between moderate and average) for the first assessment teat and 3.4 (between average and high level), while the mean score for test 1 and test 2 were 48 and 70 respec statistically significant difference between the two sets of p<0.05. The students' knowledge level increased as they made use teaching be a Key

Dokumen ini dibuat untuk kepentingan pribadi dan golongan, sehingga segala kesalahan dan ‘ketersesatan’ yang diakibatkan oleh penggunaan dokumen ini, penulis tidak bertanggung jawab. Ketersediaannya di internet bukan berarti ditujukan untuk penggunaan umum, ketersediaan tersebut dimaksudkan sebagai dokumentasi on‐line yang dimiliki penulis dan diperbolehkan dimiliki oleh siapa saja. Tulisan ini merupakan salinan ulang dari catatan perkuliahan Analisis Real di Institut Pertanian Bogor Sekolah
Pascasarjana Program Magister Sains Mayor Matematika Terapan yang diampu oleh Bapak Dr. Jaharuddin, MS dan Ibu Berlian (Bu Anggi)

Outline:
1. Teori Himpunan
2. Sistem Bilangan Real
3. Ukuran Lebesgue
4. Integral Lebesgue
5. Turunan dan Integral
6. Ruang Banach

Referensi:
1. H.L. Royden, 1988, Real Analysis, Mc. Milan Pub, New York
2. Bartle, R.G., & D.R. Sherbert, 2000, Introduction to Real Analysis, John Wiley, New York
3. Goldberg, Richard R., 1976, Methods of Real Analysis, John Wiley, New York

Pada skripsi ini dibahas dan dibuktikan sifat-sifat dari d-subaljabar fuzzy dan d-ideal fuzzy pada d-aljabar, diantaranya hubungan antara level subalgebra dengan d-subaljabar fuzzy. Selain itu, dibahas juga hubungan antara hasil kali kartesius dengan d-ideal fuzzy dan hubungan antara relasi fuzzy terkuat dengan d-ideal fuzzy

Let f : R + → R + and (an) ∞ n=1 be a sequence of positive reals. We will say that (an) ∞ n=1 is relatively (R)-dense for f provided that for every x, y ∈ R + with f (x) < f (y) there exists n, m ∈ N such that f (x) < f (an) f (am) < f (y). Sufficient conditions are given for a sequence of positive reals to be relatively (R)-dense for certain functions.

In this short note we prove with elementary techniques that the sequence xn = n k=1 n n 2 +k 2 is increasing and its limit is π 4. Moreover, we give a sufficient condition for the monotonicity of some Riemann-type sums assigned to uniform subdivisions as a function of the number of the intervals from the subdivision. This mathematical content came up in a group discussion during an IBL centered teacher training activity and reflects a crucial problem is implementing IBL teaching attitudes in the framework of a highly scientific curricula (such as the Romanian mathematics curricula for upper secondary school).

Sistem matematika (A, ∧, ∨, (•) ∼) adalah Pra A*-Aljabar, bila anggotaanggotanya memenuhi sifat-sifat tertentu. Sistem (A, ∧, ∨, (•) ∼) ditulis A yang menyatakan Pra A*-Aljabar. Misalkan didefinisikan sebuah relasi terurut parsial " " pada Pra A*-Aljabar (yang anggota-anggotanya memenuhi sifat refleksif, antisimetri, dan transitif). Kemudian x y jika dan hanya jika y ∧ x = x ∧ y = x. Himpunan A bersamasama dengan relasi terurut parsial pada A dinamakan dengan poset. Pada tulisan ini dikaji struktur aljabar dari Pra A*-Aljabar. Selanjutnya, juga dikaji sifat-sifat Pra A*-Aljabar sebagai sebuah poset.

During the nineteenth century, building on the work begun in 1821 by Augustin-Louis Cauchy (1789-1857) in his Cours d'analyse, the calculus was given a rigorous foundation that is still accepted today. By the middle of the century, developments in subject persuaded many mathematicians that the traditional concepts of the calculus were too imprecise, unreliable, and ineffective to provide such a basis. It was held that the traditional relation between real quantities and intuitively given continuous magnitudes such as straight lines was more of a hindrance than an aid in achieving that end as was the then familiar reliance on infinitesimals. In response, the modern arithmetico-set-theoretic conception of a real number emerged when a number of mathematicians including Cantor (1872/1939) and Dedekind (1872/1996) introduced systems of real numbers that were designed to dispense with the former and provide a basis for the calculus that made superfluous the latter. Cantor's system is based on Cauchy sequences of rational numbers. A sequence {r n } of rational numbers (indexed over the natural numbers) is said to be a Cauchy

It will be shown that a combination of the Axiom of Choice and Nested Intervals Property contradicts the diagonal argument or non-denumerability of real numbers. To show that, we start with a set of all algebraic numbers S and define a function f , which inserts a transcendental number between every pair of algebraic numbers within S. To show that such an expanded set denoted S , which in addition to algebraic numbers includes transcendental numbers generated by the function f , contains all real numbers used is a counterexample. Defined is an arbitrarily chosen missing number T x , not included in set S , with a sequence of algebraic pairs (intervals with algebraic endpoints) converging to T x. Then, AC is used to show that exactly the same sequence of closed intervals that defines the number T x exists in the set S. Subsequently, the Nested Intervals Property is applied to show that the recognized sequence of closed intervals within S has a nonempty intersection and contains the number T x. Given that number T x is chosen arbitrarily and is in set S , it implies that all numbers are in the set S , which is also obviously denumerable.

I was flipping through a Calculus text yesterday and as I was glancing through the topics, the alternating series test caught my attention. I started to think about how the proof would go. So I set out to write out the proof and here it is in case you don't want to pull out a Calculus textbook to find out or you need an extra reference.

It is well known that the harmonic series diverges because its sequence of partial sums is not bounded. The aim of this article is to show how slow this divergence is. For this purpose, we use a simple theorem that all undergraduate students are familiar with. Let us start with the theorem used very often to check various series for convergence. T heorem 1 I f (an)^Lj is a decreasing sequence o f positive real numbers, then the series 5 1 ^ = 1 °n converges i f and only i f the series 2 na 2&quot; converges. Proof. Let s n denotes the n-th partial sum of the series on, i.e. = oi + 0 2 + • • + on and let tn denotes the n-th partial sum of the series 2 n0 2 « , i.e. tn — 2^ 0 2 0 + 2 , o2. + . + 2 n0 2 &quot; ■ Since an &gt; 0 for all n both the partial sums s n and the partial sums tn form an increasing sequence. From S o * — S 2 * i — O i + G 2 + ■ • + 0 2 * 1 + O j * + • + 4*2* — (Oi + 0,2 + . . . + fl2fc_1 ) = 02*-! “I&quot; 02*-i-i-i + . . . + C2* we infer 2*l a2* ^ S2* — S2*-i $...

A. Bromaıı generaliscd Holditeh's Theorem to the closed reetifiable curves [2 ]. In this paper we obtaîned an interesting expression, which is similar to Holditeh's Formula, for the polar inertia momentums of the closed reetifiable curves and we show that the ratio between the polar inertia momentums of the curves and the areas bounded by them is 1 / 2.

Beberapa kekurangan dari konsep jumlah Riemann diatasi dengan cara memperumum konsep tersebut. Dalam perumuman ini digunakan konsep mengenai ukuran dan himpunan terukur ; pada himpunan-himpunan tak kosong di Rn, dikenal ukuran khusus yang disebut dengan ukuran Lebesgue, yang diperoleh dari pembatasan ukuran luar Lebesgue pada koleksi dari semua himpunan (terukur) Lebesgue. Ukuran luar Lebesgue juga digunakan dalam pemverifikasian apakah suatu subhimpunan di Rn merupakan himpunan Lebesgue. Dalam tulisan ini diperkenalkan dan dijelaskan tentang sebuah metode yang dapat digunakan untuk mengonstruksi suatu ukuran luar, bentuk pendefinisian ukuran luar Lebesgue, beserta sifat-sifat dari ukuran luar Lebesgue, yang didasarkan kepada definisi ukuran luar dan sebuah teorema tentang pengonstruksian suatu ukuran luar. Kata Kunci: ukuran luar Lebesgue, himpunan Lebesgue, ukuran Lebesgue

Sequence spaces is one of interesting topic of analysis research. Some of that are convergent and bounded sequence. In this paper we define some releated BK spaces  (????,?????)?, (????,?????), dan (????,?????)0 some characteristics releated to them. Moreover we determine  their ?-dual.

The characteristics of a function are usually investigated by looking at the continuity of the function. But what happens if a function does not have continuous properties? To what extent can the characteristics of continuous function be maintained for discontinuous cases? The stochastic function that is widely involved in solving problems in the field of average financial mathematics is a discontinuous function. This is reflected by the acquisition of a smooth curve from the modeling drawing obtained. Today, the nature of continuous functions in [a, b] has been widely studied and developed. Some properties of the continuous function can be extended to the appropriate discontinuous function. In this paper, there will be some integral reviews for discontinuous functions which are closely related to stochastic functions.

The space l^p with 1≤p∞ is the set of real numbers that satisfy _(n=1)^∞▒〖|x_n |^p∞〗.The function in the vector space X which has real value which fulfills the norm-2 properties is denoted by ,⋅‖ and the pair (X,‖⋅,⋅‖) is called the norm-2 space.A norm-2 space is said to be complete or called a Banach-2 space if every Cauchy sequence in the space converges to an element in that space.This research was conducted to prove the l^p space in the complete norm-2.The first step to prove the completeness is to prove that the norm contained in l^p with 1≤p∞ satisfies the properties of norm-2.Next, prove that the norm derived from norm-2 is equivalent to the norm in l^p.Next shows that every Cauchy sequence in space l^p converges to an element in space l^p.Based on this proof, it is found that (l^p,‖⋅,⋅‖) is a complete norm-2 space.

Riemann-Stieltjes Integration Calculus provides us with tools to study nicely behaved phenomena using small discrete increments for information collection. The general idea is to (intelligently) connect information obtained from examination of a phenomenon over a lot of tiny discrete increments of some related quantity to "close in on" or approximate something that behaves in a controlled (i.e., bounded, continuous, etc.) way. The "closing in on" approach is useful only if we can get back to information concerning the phenomena that was originally under study. The bene¿t of this approach is most beautifully illustrated with the elementary theory of integral calculus over U. It enables us to adapt some "limiting" formulas that relate quantities of physical interest to study more realistic situations involving the quantities. Consider three formulas that are encountered frequently in most standard physical science and physics classes at the pre-college level:

Pada tahun 1875 G. Darboux memodifikasi definisi Integral Riemann dengan terlebih dahulu mendefinisikan jumlah Darboux atas dan Darboux bawah, selajutnya mendefinisikan Integral Darboux atas dan Integral Darboux bawah. Keduanya memiliki ekuivalensi yaitu ... Suatu fungsi dikatakan terintegral Darboux jika dan hanya jika ia juga merupakan terintegral Riemann, dan jika nilai-nilai Integral dari keduanya ada, maka bersifat sama. Integral Darboux mempunyai keuntungan lebih sederhana dibanding Integral Rieman. Karena ekuivalensi maka sifat Integral Riemann yakni ketunggalan nilai, kelinieran, keterbatasan juga berlaku pada Integral Darboux. Adapun sifatnya adalah:...

Abstrak: Tulisan ini menyajikan definisi dan teorema limit fungsi dan limit barisan pada topologi real yang bertujuan untuk mengetahui hubungan antara limit fungsi dan limit barisan pada topologi real. Misalkan diberikan suatu barisan pada bilangan real dan suatu bilangan real , bilangan adalah limit barisan jika untuk setiap lingkungan dari , barisan adalah eventually pada , dan Misalkan , dengan suatu bilangan real, (himpunan semua titik limit dari). Bilangan dikatakan limit dari. Jika untuk setiap lingkungan dari mendapatkan lingkungan dari sedemikian sehingga untuk setiap. Akan mencari hubungan antara kedunya, yaitu dengan menggunakan definisi dan teorema mengenai limit fungsi dan limit barisan pada topologi real.

In this paper, we will try to proof existence of supremum and infimum with Cantor Dedekind theorem. Before we discuss this material, necessary to introduce several basic concepts, especially Cut Dedekind and Supremum and Infimum concepts. The method we are presenting here is due to Richard Dedekind (1831-1916) whose work entitled "What are and what should be numbers?". Cantor Dedekind theorem very important to show that nothing "gap" at real numbers system.

We determine sufficient conditions on positive weights W and V such that there exists continuous, strictly increasing functions 8 and 9 on [0,) such that 8(0)= 0=9(0) and 8 \ &Wf & | f (0)| + 9 &Vf & | f (0)| + 1 whenever f: R Ä R is a continuous integrable function. We also give an example that shows the optimality of our conditions.

INDONESIA: Analisis neo-klasik merupakan sintesis analisis klasik, teori himpunan fuzzy dan analisis himpunan nilai. Pada dasarnya, bentuk analisisnya sederhana, seperti fungsi-fungsi dan operasi-operasi yang telah dipelajari berdasarkan pengertian konsep fuzzy : limit fuzzy, kekontinyuan fuzzy, dan turunan fuzzy. Oleh karena itu, butuh metode-metode baru untuk menguraikan ketaksamaan. Untuk mencapai tujuan tersebut, konsep limit diperluas pada konsep limit fuzzy atau r-limit. Penelitian ini bertujuan untuk menjelaskan sifat-sifat limit fuzzy dari suatu fungsi di R+. Pembahasan mengenai limit fuzzy dari suatu fungsi, awalnya, mengembangkan dan menunjukkan konstruksi limit fuzzy dari fungsi yang hampir mirip dengan limit fuzzy dari barisan. Oleh karena itu, limit fuzzy dari fungsi ini tingkatannya lebih tinggi dari konsep klasik limit fungsi. Pendefinisian r-limit dari fungsi f(x) di titik berdasar pada konsep r-limit barisan. Barisan yang digunakan yaitu barisan yang konvergen. Pada...

El Cálculo (Cálculo Infinitesimal) es una de las más hermosas y útiles ramas de la ciencia matemática. Algunos excelentes textos de  Precálculo,  Cálculo Diferencial y de Cálculo Integral, en una o varias variables, son los siguientes: (1) "PRECÁLCULO (Matemáticas para el Cálculo)" de Stewart, Redlin y Watson, (2) "EL CÁLCULO" de Leithold, (3) "CÁLCULO, una variable" de Thomas, (4) CÁLCULO, varias variables" de Thomas. (5) "CÁLCULO, TRASCENDENTES TEMPRANAS", de Stewart,  (6) "CÁLCULO DIFERENCIAL, con funciones trascendentes tempranas, para ciencias e ingeniería" de Saenz, (7) "CÁLCULO INTEGRAL, con funciones trascendentes tempranas, para ciencias e ingeniería" de Saenz,  (8) "CÁLCULO INFINITESIMAL", de  Spivak, y (9) "CALCULUS" de  Apostol. Todos ellos se pueden encontrar y bajar en varios sitios de la web, y en particular, en la biblioteca digital de mi blog "MATEMÁTICAS: PURAS y APLICADAS", enlace: http://logicamatematica-lm.blogspot.com/
El siguiente pequeño documento (Nota) que se anexa es una página PDF que contiene una  imagen de cada uno de ellos (de una de sus ediciones), dicha bibliografía puede ser útil para aprender y enseñar  tal ciencia matemática.

Berisi makalah terkait materi Nilai Mutak, Kelengkapan Bilangan R, Ketaksamaan Segitiga, Supremum Infimum, dan Hukum Archimedes. Makalah di tulis diadopsi dari terjemahan buku Analisis Riil Bartle yang kemudian di jabarkan dalam membuktikan teorema-teorema maupun definisi di Buku tersebut. Disertai juga latihan soal dan jawaban.

The inspiration of the definition of &quot;compactness&quot; comes from the real number system. Closed and bounded sets in the real line were considered as an excellent model to show a generalized version of the compactness in a topological space. Since boundedness is an elusive concept in general topo-logical space, then the compact properties are analysed to look at some properties of sets that do not use boundedness. Some of the classical results of this nature are Bolzano -Weierstrass theorem, whe-re every infinite subset of [a,b] has accumulation point and Heine-Borel theorem, where every closed and bounded interval [a,b] is compact. Each of these properties and some others are used to define a generalized version of compactness. Hausdorff space has compact properties if every compact subset in Hausdorff space is closed and every infinite Hausdorff space has infinite sequence of non empty and disjoint open sets. Because the compact properties in the Hausdorff space are satisfie...

Se verá las reglas de integrabilidad para el manejo eficiente de las integrales
viendo los teoremas mas importantes así como la forma integral del resto del teorema de Taylor e integrales de funciones logarítmicas, exponenciales e impropias.

Ruang vektor yang dilengkapi dengan aksioma inner product disebut ruang inner product (pre-Hilbert). Ruang pre-Hilbert dikatakan lengkap jika setiap barisan Cauchy di dalamnya konvergen. Ruang pre-Hilbert yang lengkap adalah ruang Hilbert. Diberikan ruang fungsi   L^2 adalah himpunan semua fungsi bernilai kompleks yang mempunyai integral mutlak kuadrat berhingga  dan  merupakan suatu ruang vektor. Ruang fungsi L^2 yang dilengkapi inner product  membentuk ruang pre-Hilbert. Dalam penelitian ini ditunjukkan bahwa ruang fungsi  tersebut merupakan ruang Hilbert. Dari sifat kelengkapan dapat ditunjukkan setiap barisan Cauchy  di dalam ruang fungsi  konvergen maka ruang fungsi merupakan  ruang Hilbert.Kata kunci: Pre-Hilbert, Hilbert, ruang fungsi .