CSIR NET Mathematical Science UNIT – 1 Syllabus

UNIT – 1 syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for UNIT – 1 in CSIR NET Mathematical Science, not a summary of it.

1. 1

   Analysis

   15 topics

   • Elementary set theory
   • Finite, countable and uncountable sets
   • Real number system as a complete ordered field
     • Archimedean property
     • supremum
     • infimum
   • Sequences and series
     • convergence
     • limsup
     • liminf
   • Bolzano Weierstrass theorem
   • Heine Borel theorem
   • Continuity
     • uniform continuity
   • Differentiability
     • mean value theorem
   • Sequences and series of functions
     • uniform convergence
   • Riemann sums and Riemann integral
     • Improper Integrals
   • Monotonic functions
   • Types of discontinuity
   • Functions of bounded variation
   • Lebesgue measure
   • Lebesgue integral
2. 2

   Linear Algebra

   6 topics

   • Vector spaces
     • subspaces
     • linear dependence
     • basis
     • dimension
     • algebra of linear transformations
   • Algebra of matrices
     • rank and determinant of matrices
     • linear equations
   • Eigenvalues and eigenvectors
     • Cayley-Hamilton theorem
   • Matrix representation of linear transformations
     • Change of basis
     • canonical forms
     • diagonal forms
     • triangular forms
     • Jordan forms
   • Inner product spaces
     • orthonormal basis
   • Quadratic forms
     • reduction and classification of quadratic forms

UNIT – 1 flashcards for CSIR NET Mathematical Science

23 of 51 cards from the UNIT – 1 deck — real questions with worked answers.

1. In elementary set theory, state De Morgan's laws for the complement of a union and an intersection of sets $A$ and $B$.

   $(A \cup B)^{c} = A^{c} \cap B^{c}$ and $(A \cap B)^{c} = A^{c} \cup B^{c}$.

2. Define a countable set and distinguish countably infinite from finite.

   A set is countable if it is finite or there is a bijection with $\mathbb{N}$. If the bijection with $\mathbb{N}$ exists it is countably infinite; otherwise (a finite bijection with $\{1,\dots,n\}$) it is finite.

3. Is the set of rational numbers $\mathbb{Q}$ countable or uncountable, and what is a key consequence for $\mathbb{R}$?

   $\mathbb{Q}$ is countable. Since $\mathbb{R}$ is uncountable, the irrationals $\mathbb{R}\setminus\mathbb{Q}$ are uncountable.

4. State Cantor's diagonal argument conclusion about the interval $[0,1]$.

   The interval $[0,1]$ (and hence $\mathbb{R}$) is uncountable: no surjection $f:\mathbb{N}\to[0,1]$ exists, shown by constructing a number differing from $f(n)$ in its $n$-th decimal digit.

5. Give the cardinality relationship (Cantor's theorem) between a set $A$ and its power set $\mathcal{P}(A)$.

   $|A| < |\mathcal{P}(A)|$ always; there is no surjection from $A$ onto $\mathcal{P}(A)$. For finite $A$, $|\mathcal{P}(A)| = 2^{|A|}$.

6. Is a countable union of countable sets countable?

   Yes. A countable union of countable sets is countable (using the axiom of choice).

7. What does it mean that $\mathbb{R}$ is a complete ordered field?

   $\mathbb{R}$ is a field with a compatible total order, and it satisfies the completeness (least upper bound) axiom: every nonempty subset bounded above has a supremum in $\mathbb{R}$.

8. Define the supremum (least upper bound) of a nonempty set $S \subseteq \mathbb{R}$ bounded above.

   $\sup S = M$ where (i) $M$ is an upper bound: $x \leq M$ for all $x \in S$, and (ii) $M$ is least: for every $\varepsilon > 0$ there exists $x \in S$ with $x > M - \varepsilon$.

9. Define the infimum (greatest lower bound) of a nonempty set $S \subseteq \mathbb{R}$ bounded below.

   $\inf S = m$ where (i) $m$ is a lower bound: $x \geq m$ for all $x \in S$, and (ii) for every $\varepsilon > 0$ there exists $x \in S$ with $x < m + \varepsilon$.

10. State the Archimedean property of $\mathbb{R}$.

    For any real numbers $x > 0$ and $y$, there exists $n \in \mathbb{N}$ such that $nx > y$. Equivalently, $\mathbb{N}$ is unbounded above and $\inf\{1/n : n \in \mathbb{N}\} = 0$.

11. What property of $\mathbb{Q}$ in $\mathbb{R}$ follows from the Archimedean property?

    Density: between any two reals $a < b$ there exists a rational number $q$ with $a < q < b$ (and also an irrational).

12. Define convergence of a sequence $(a_n)$ to a limit $L$ (epsilon definition).

    $a_n \to L$ means: for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that for all $n \geq N$, $|a_n - L| < \varepsilon$.

13. State the Monotone Convergence Theorem for sequences.

    Every monotone (increasing or decreasing) sequence that is bounded converges. An increasing bounded sequence converges to its supremum; a decreasing bounded sequence to its infimum.

14. Define a Cauchy sequence and state its relation to convergence in $\mathbb{R}$.

    $(a_n)$ is Cauchy if for every $\varepsilon > 0$ there is $N$ with $|a_m - a_n| < \varepsilon$ for all $m,n \geq N$. In $\mathbb{R}$ (complete), a sequence converges iff it is Cauchy.

15. Define $\limsup_{n\to\infty} a_n$ for a real sequence.

    $\limsup_{n\to\infty} a_n = \lim_{n\to\infty}\Big(\sup_{k \geq n} a_k\Big) = \inf_{n}\sup_{k\geq n} a_k$, the largest subsequential limit.

16. Define $\liminf_{n\to\infty} a_n$ for a real sequence.

    $\liminf_{n\to\infty} a_n = \lim_{n\to\infty}\Big(\inf_{k \geq n} a_k\Big) = \sup_{n}\inf_{k\geq n} a_k$, the smallest subsequential limit.

17. State the criterion relating $\limsup$, $\liminf$ to convergence.

    A sequence $(a_n)$ converges to $L$ (in the extended reals) iff $\liminf a_n = \limsup a_n = L$. Always $\liminf a_n \leq \limsup a_n$.

18. State the Bolzano-Weierstrass theorem.

    Every bounded sequence in $\mathbb{R}$ (or $\mathbb{R}^n$) has a convergent subsequence. Equivalently, every bounded infinite subset of $\mathbb{R}$ has a limit point.

19. State the Heine-Borel theorem for $\mathbb{R}^n$.

    A subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded.

20. State the $n$-th term test (divergence test) for a series $\sum a_n$.

    If $\sum a_n$ converges then $a_n \to 0$. Contrapositive: if $a_n \not\to 0$, the series diverges. (The converse is false, e.g. $\sum 1/n$.)

21. State the Ratio Test for the series $\sum a_n$.

    Let $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$. If $L < 1$ the series converges absolutely; if $L > 1$ (or $\infty$) it diverges; if $L = 1$ the test is inconclusive.

22. State the Root Test for the series $\sum a_n$.

    Let $L = \limsup_{n\to\infty}\sqrt[n]{|a_n|}$. If $L < 1$ converges absolutely; if $L > 1$ diverges; if $L = 1$ inconclusive.

23. For which values of $p$ does the $p$-series $\sum_{n=1}^{\infty} \frac{1}{n^{p}}$ converge?

    It converges if and only if $p > 1$, and diverges for $p \leq 1$.

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