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- MTH321: Real Analysis I (Spring 2023) @atiq
- erges to $s$. - For each irrational number $x$, there exists a sequence $\left\{ {{r}_{n}} \right\}$ of... and only if for any real number $\varepsilon >0$, there exists a positive integer ${{n}_{0}}$ such that $... r $\lambda $ that lies between $f(a)$ and $f(b)$, there exist a point $c\in (a,b)$ with $f(c)=\lambda $. ... us on $[a,b]$and differentiable on $(a,b)$. Then there exists a point $c\in (a,b)$ such that $\frac{f(b
- MTH322: Real Analysis II (Spring 2023) @atiq
- {\,\infty }{f(x)\,dx}$ converges if, and only if, there exists a constant $M>0$ such tha $\int\limits_{a}... or every $b\ge a$ and for every $\varepsilon >0\,$there exists a $B>0\,$ such that \[\left| \,\int_{b}^{... or every $\varepsilon>0$ and for all $x\in[a,b]$, there exist an integer $N$ such that $$\left|f_{n+p}(x)... converge uniformly (and absolutely) on $[a,b]$ if there exists a convergent series $\sum M_n$ of positive
- Question 7 and 8, Exercise 6.3 @math-11-nbf:sol:unit06
- has exactly $2$ women. ** Solution. ** (i) If there are exactly $2$ women then there will be $3$ man in committee. Total possible ways $={ }^{4} C_{2} \tim... women. ** Solution. ** At least $2$ women means there could be more than $2$ women as well.\\ So we wil... having $2,3$ and four women in committee.\\ So if there are $2$ women we already had calculated $120$ pos
- Real Analysis: Short Questions and MCQs @msc:mcqs_short_questions
- e between rational and irrational numbers? - Is there a rational number exists between any two rational numbers. - Is there a real number exists between any two real numbers... ><panel> 8. A set $A$ is said to be countable if there exists a function $f:A\to \mathbb{N}$ such that ... ence $\{s_n\}$ is said to be bounded if * (A) there exists number $\lambda$ such that $|s_n|<\lambda
- MTH321: Real Analysis I (Fall 2021) @atiq
- is for modern decimal notation, and insisted that there is no difference between rational and irrational ... iles of the notes given below. To view PDF files, there must be PDF Reader (Viewer) installed on your PC ... * 1.19- Given two real numbers $x$ and $y$, $x<y$ there is an irrational number $u$ such that $x<u<y$. * 1.20- For every real number $x$ there is a set $E$ of rational number such that $x=\sup
- MCQs or Short Questions @atiq:sp15-mth321
- \sqrt{2}$ * (C) $\sqrt{3}$ * (D) 7 - Is there a rational number exists between any two rational numbers. - Is there a real number exists between any two real numbers... untable? - A set $A$ is said to be countable if there exists a function $f:A\to \mathbb{N}$ such that ... ence $\{s_n\}$ is said to be bounded if * (A) there exists number $\lambda$ such that $|s_n|<\lambda
- Chapter 03 - Limits and Continuity @msc:real_analysis_notes_by_syed_gul_shah
- if for every positive real number $\varepsilon$, there is $\delta>0$ such that $|f(t)-f(s)|<\varepsilon$... on //X//. If //f// is continuous at $c\in X$ then there exists a number $\delta>0$ such that //f// is bou... on [a, b]. If $f(c)>0$ for some $c\in [a,b]$ then there exist an open interval $G \subset[a,b]$ such that... n [//a//,//b//] (i) If $f(a)<0$ and $f(b)>0$ then there is a point //c//, $a<c<b$ such that $f(c)=0$. (ii
- Question 9 Exercise 6.3 @math-11-kpk:sol:unit06
- . In how many ways can the committee be chosen if there must be at least two men? ====Solution==== Total ... $. Therefore, Total number of persons $=7+6=13$ There must be at least two men The number of committees... . In how many ways can the committee be chosen if there must be at least two women? ====Solution==== Tota... .$ Therefore, Total number of persons $=7+6=13$ There must be at least two women. The number of commit
- MTH321: Real Analysis I (Spring 2020) @atiq
- is for modern decimal notation, and insisted that there is no difference between rational and irrational ... iles of the notes given below. To view PDF files, there must be PDF Reader (Viewer) installed on your PC ... $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $|s_n|>... $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $|s_n|>
- History of Mathematics by Muhammad Usman Hamid @notes
- political and military events are always chaotic; there is no way to predict the rise of a Genghis Khan, ... chaotically—changed in the subsequent millennia, there are elements of greatness in all the fashions. Si... ble, or repulsive. But only among the sciences is there true progress; only there is the record one of continuous advance toward ever greater heights. And yet,
- Chapter 01 - Real Number System @msc:real_analysis_notes_by_syed_gul_shah
- ====== ==== Contents & Summary ==== * Theorem: There is no rational //p// such that $p^2=2$. * Theor... imedean property (b) Between any two real numbers there exits a rational number. * Theorem: Given two real numbers //x// and //y//, $x<y$ there is an irrational number //u// such that $x<u<y$. * Theorem: For every real number //x// there is a set //E// of rational number such that $x=\s
- Syllabus for PU @msc:syllabus
- sity of the Punjab, Lahore for paper pattern that there is a conflict on information given on University ... ed by M.Sc. (Mathematics) regular students * i. There are a total of 1200 marks for M.Sc. (Mathematics)... as is the case with other M.Sc. subjects. * ii. There are five papers in Part-I and six papers in Part-II. Each paper carries 100 marks. * iii. There is a Viva Voce Examination of M.Sc. Part II. The
- Question 23 and 24, Exercise 4.3 @math-11-nbf:sol:unit04
- rchers are in the last row? How many marchers are there altogether? ** Solution. ** From the statement,... $62$ marchers in last row and $950$ marchers are there altogether. GOOD m( =====Question 24===== How m... any poles will be in a pile of telephone poles if there are 50 in the first layer, 49 in the second and so on, until there are 6 in the last layer? ** Solution. ** From t
- Question 5 and 6, Exercise 6.3 @math-11-nbf:sol:unit06
- ny ways can $11$ players be chosen out of $16$ if there is no restriction. ** Solution. ** $11$ players... in ${ }^{16} C_{11}$ ways.\\ i.e. $4368$ ways are there to choose $11$ players.\\ =====Question 5(ii)===... ? ** Solution. ** Case I: If one man is chosen there will be two women in committee.\\ $1$ man may be ... \times{ }^{3} C_{2}=5 \times 3=15$\\ Case II: If there are $2$ men and one woman in committee. Total po
- MTH322: Real Analysis II (Fall 2021) @atiq
- a}^{\infty }{f(x) dx}$ converges if, and only if, there exists a constant $M>0$ such that $\int\limits_{a... erges if, and only if, for every $\varepsilon >0$ there exists a $B>0$ such that $c>b>B$ implies $\left| ... or every $b\ge a$ and for every $\varepsilon >0\,$there exists a $B>0\,$ such that \[\left| \,\int_{b}^{... he interval $[-A,A]$, where $A>0$. - Prove that there exists a function $E:\mathbb{R} \to \mathbb{R}$ s
- Recent Advances in Mathematical Methods, Models & Applications, LSC Lahore, Pakistan (April 13-14, 2019) @conferences
- Second Conference on Mathematical Sciences (SCMS-2013), International Islamic University, Islamabad, Pakistan (1-2 November 2013) @conferences
- Chapter 03: PDF Viewer @bsc:notes_of_calculus_with_analytic_geometry:ch03_general_theorem_intermediate_forms
- Chapter 08: PDF Viewer @bsc:notes_of_calculus_with_analytic_geometry:ch08_analytic_geometry_of_three_dimensions
- Chapter 09: PDF Viewer @bsc:notes_of_calculus_with_analytic_geometry:ch09_functions_of_several_variables
- Syllabus & Paper Pattern for General Mathematics (Split Program) @bsc:paper_pattern:punjab_university
- Ch 07: Permutation, Combination and Probability: Mathematics FSc Part 1 @fsc:fsc_part_1_solutions:ch07
- Ch 08: Mathematical Induction and Binomial Theorem: Mathematics FSc Part 1 @fsc:fsc_part_1_solutions:ch08