Foundations

A set is a well-defined collection of distinct objects, called elements or members. Sets are usually denoted by capital letters, and elements by lowercase letters. We write \( x \in A \) to mean "\( x \) belongs to set \( A \)".

Hinglish: Set matlab ek thaili (bag) jisme kuch cheezein hain. "Well-defined" ka matlab hai ki kisi bhi cheez ke baare mein clearly bata sako ki wo thaili mein hai ya nahi. Example: \( A = \{1, 2, 3\} \) — yahan 2 \( \in A \) hai, lekin 5 \( \notin A \).

\( A \subseteq B \) means every element of \( A \) is also an element of \( B \). We say "\( A \) is a subset of \( B \)". If \( A \neq B \), we call it a proper subset (\( A \subset B \)).

Hinglish: Agar thaili A ki har ek cheez thaili B mein bhi mil jaaye, toh A, B ka subset hai. Example: \( \{1, 2\} \subseteq \{1, 2, 3\} \) ✅ lekin \( \{1, 4\} \nsubseteq \{1, 2, 3\} \) ❌ (kyunki 4 nahi hai B mein).

\( A \cup B \) (Union) = all elements in \( A \) or \( B \) (ya dono). \( A \cap B \) (Intersection) = all elements in both \( A \) and \( B \). \( A^c \) (Complement) = everything not in \( A \).

Hinglish: Union (∪) = dono thailiyon ki cheezein mila do. Intersection (∩) = sirf wahi cheezein jo dono mein hon. Complement = jo cheez set mein nahi hai. Example: \( \{1,2\} \cup \{2,3\} = \{1,2,3\} \), \( \{1,2\} \cap \{2,3\} = \{2\} \).

A function \( f: A \to B \) is a rule that assigns to each element \( x \in A \) (called the domain) exactly one element \( f(x) \in B \) (called the codomain). The set of all actual outputs is the range (or image).

Hinglish: Function ek machine hai — tum input daalo (domain se), machine ek output degi (codomain mein). Har input ka exactly ek output hona chahiye. Example: \( f(x) = x^2 \). Input \( 3 \) daalo → output \( 9 \). Input \( -2 \) → output \( 4 \).

A field is a set \( F \) equipped with two operations (addition + and multiplication ×) such that: (1) \( (F, +) \) is a commutative group with identity 0, (2) \( (F \setminus \{0\}, \times) \) is a commutative group with identity 1, and (3) multiplication distributes over addition. In simple words: you can add, subtract, multiply, and divide (except by 0) and everything behaves "normally".

Hinglish: Field matlab ek aisa number system jismein jod, ghata, guna, aur bhaag (÷ except 0) — charon kaam freely ho sakein. Example: \( \mathbb{Q} \) (rationals) ek field hai kyunki 3/4 ÷ 2/5 = 15/8, jo ab bhi rational hai. \( \mathbb{Z} \) (integers) field nahi hai kyunki 1 ÷ 2 = 0.5, jo integer nahi hai!

An ordered field is a field with a total order relation \( < \) that is compatible with field operations: (1) If \( a < b \), then \( a + c < b + c \). (2) If \( a < b \) and \( c > 0 \), then \( ac < bc \).

Hinglish: Ordered field = field + "bada-chhota" ka concept. \( \mathbb{R} \) ordered field hai — tum kisi bhi do numbers ko compare kar sakte ho (3 < 5). Lekin \( \mathbb{C} \) ordered field nahi hai — kya \( 3 + 2i \) bada hai ya \( 1 + 4i \)? Koi consistent tarika nahi hai decide karne ka!

A number \( M \) is an upper bound of a set \( A \) if \( a \leq M \) for every \( a \in A \). The set is then called bounded above.

Hinglish: Upper bound matlab ek aisi "chhat" (ceiling) ki set ka koi bhi element us se oopar nahi jaa sakta. Example: Set \( A = \{1, 3, 5\} \) ke liye 5, 6, 7, 100, 999 — sab upper bounds hain. Sabse chhota upper bound (= 5) ko supremum kehte hain.

A number \( m \) is a lower bound of \( A \) if \( m \leq a \) for every \( a \in A \). The set is then called bounded below.

Hinglish: Lower bound matlab ek "farsh" (floor) — set ka koi element us se neeche nahi jaa sakta. Example: Set \( A = \{1, 3, 5\} \) ke liye 1, 0, -5, -100 sab lower bounds hain. Sabse bada lower bound (= 1) ko infimum kehte hain.

\( s = \sup A \) if (i) \( s \) is an upper bound of \( A \), and (ii) if \( u \) is any other upper bound, then \( s \leq u \). In other words, \( \sup A \) is the smallest among all upper bounds.

Hinglish: Bohot saari "chhatein" hain set ke oopar — sabse neechi chhat ko supremum kehte hain. Example: \( A = (0, 1) = \{x : 0 < x < 1\} \). Upper bounds: 1, 2, 3, 100... Sabse chhota upper bound = \( \sup A = 1 \). Note: 1 khud set mein nahi hai!

\( l = \inf A \) if (i) \( l \) is a lower bound of \( A \), and (ii) if \( m \) is any other lower bound, then \( m \leq l \). In other words, \( \inf A \) is the largest among all lower bounds.

Hinglish: Bohot saare "farsh" hain set ke neeche — sabse ooncha farsh ko infimum kehte hain. Example: \( A = (0, 1) \). Lower bounds: 0, -1, -100... Sabse bada lower bound = \( \inf A = 0 \). Yahan bhi 0 set mein nahi hai!

\( \max A \) exists if and only if \( \sup A \in A \) (the supremum is actually inside the set). Similarly, \( \min A \) exists iff \( \inf A \in A \).

Hinglish: Max = sup + wo element set ke andar bhi ho. Example: \( A = [0, 1] \) → \( \max A = 1 \) (kyunki 1 \( \in A \)). Lekin \( A = (0, 1) \) → \( \max A \) exist hi nahi karta (1 set mein nahi hai, bhale hi sup = 1 hai)!

A set \( A \subseteq \mathbb{R} \) is bounded if it is both bounded above and bounded below, i.e., there exist \( m, M \in \mathbb{R} \) such that \( m \leq a \leq M \) for all \( a \in A \). Equivalently, \( A \) fits inside some interval \( [m, M] \).

Hinglish: Bounded = set ek dabba (box) mein fit ho jaaye. Examples: \( (0, 1) \) bounded hai ✅ (0 aur 1 ke beech hai). \( (0, \infty) \) bounded nahi hai ❌ (infinity tak jaata hai). \( \mathbb{Z} = \{...,-2,-1,0,1,2,...\} \) bounded nahi ❌.

A sequence is a function \( x: \mathbb{N} \to \mathbb{R} \). We write \( x_n \) (or \( (x_n)_{n=1}^{\infty} \)) instead of \( x(n) \). It is an ordered list of real numbers indexed by natural numbers.

Hinglish: Sequence = numbers ki ek line (queue) jo ek pattern follow karti hai. Examples: \( x_n = 1/n \) → sequence: \( 1, 1/2, 1/3, 1/4, \dots \). \( x_n = (-1)^n \) → sequence: \( -1, 1, -1, 1, \dots \) (yeh oscillate karti hai).

A sequence \( (x_n) \) converges to a limit \( L \) (written \( x_n \to L \) or \( \lim_{n \to \infty} x_n = L \)) if for every \( \varepsilon > 0 \), there exists \( N \in \mathbb{N} \) such that \( n \geq N \implies |x_n - L| < \varepsilon \).

Hinglish: Converge = sequence ek number ke itna paas aa jaaye ki practically wo wahi ho jaaye. Jaise \( 1, 0.5, 0.33, 0.25, \dots \) → converges to 0. Chahe kitna bhi tight band (\( \varepsilon \)) lagao 0 ke around, ek point ke baad saare terms us band mein aa jaayenge.

A sequence diverges if it does not converge to any finite limit. It may go to \( +\infty \), \( -\infty \), or oscillate without settling.

Hinglish: Diverge = sequence kisi ek jagah tikti nahi. Examples: \( x_n = n \) → \( 1, 2, 3, 4, \dots \) (infinity ko jaata hai — diverges to \( +\infty \)). \( x_n = (-1)^n \) → \( -1, 1, -1, 1, \dots \) (baar baar uchalta hai — diverges by oscillation).

A subsequence of \( (x_n) \) is a new sequence \( (x_{n_k}) \) formed by picking elements from \( (x_n) \) at increasing indices \( n_1 < n_2 < n_3 < \dots \). You can skip terms, but you cannot reorder them.

Hinglish: Subsequence = original queue mein se kuch logon ko chun lena (lekin order wahi rehna chahiye). Example: Sequence \( 1, 2, 3, 4, 5, 6, \dots \) mein se sirf even terms chuno: \( 2, 4, 6, \dots \) — ye ek subsequence hai. Sirf prime positions chuno: \( x_2, x_3, x_5, x_7, \dots \) — ye bhi valid subsequence.

A sequence \( (x_n) \) is Cauchy if for every \( \varepsilon > 0 \), there exists \( N \) such that \( m, n \geq N \implies |x_m - x_n| < \varepsilon \). In \( \mathbb{R} \), a sequence converges if and only if it is Cauchy.

Hinglish: Cauchy sequence mein terms aapas mein itne paas aa jaate hain ki unke beech ka gap negligible ho jaata hai. Fark samjho: Convergence mein hum puchhte hain "kya terms limit L ke paas hain?" Cauchy mein puchhte hain "kya terms ek doosre ke paas hain?" \( \mathbb{R} \) mein dono ek hi baat hai! Lekin \( \mathbb{Q} \) mein Cauchy sequence converge nahi bhi ho sakti (kyunki limit irrational ho sakta hai).

An \( \varepsilon \)-neighborhood of a point \( x \) is the open interval \( V_\varepsilon(x) = (x - \varepsilon, x + \varepsilon) \) — all points within distance \( \varepsilon \) from \( x \).

Hinglish: Neighborhood = ek point ke aas paas ka chhota sa ilaaqa. Jaise tumhare ghar ke 100 meter ke andar jo bhi hai, wo tumhara "neighborhood" hai. Example: \( V_{0.5}(3) = (2.5, 3.5) \) — point 3 se 0.5 ki doori ke andar sab kuch.

A set \( U \subseteq \mathbb{R} \) is open if for every point \( x \in U \), there exists \( \varepsilon > 0 \) such that \( (x - \varepsilon, x + \varepsilon) \subseteq U \). Every point can "breathe" — it has room around it.

Hinglish: Open set = set ke har point ke charon taraf jagah hai set ke andar hi. Koi bhi point "katore ke kinaare" pe nahi hai. Example: \( (0, 1) \) open hai ✅ — 0.5 ke paas 0.49 aur 0.51 dono set mein hain. \( [0, 1] \) open nahi hai ❌ — point 0 ke left mein kuch nahi hai set mein.

A set \( F \subseteq \mathbb{R} \) is closed if it contains all of its limit points. Equivalently, its complement \( \mathbb{R} \setminus F \) is open.

Hinglish: Closed set = agar koi sequence set ke andar se aakar kisi point pe converge karti hai, toh wo point bhi set ke andar hona chahiye. Set apne "boundary" ko khud rakhta hai. Example: \( [0, 1] \) closed hai ✅ (boundary points 0 aur 1 andar hain). \( (0, 1) \) closed nahi ❌ (0 aur 1 limit points hain lekin set mein nahi).

In \( \mathbb{R} \), a set \( K \) is compact if and only if it is closed and bounded (Heine-Borel Theorem). Compact sets are "the best" sets — every sequence in them has a convergent subsequence whose limit is also in the set.

Hinglish: Compact = Closed + Bounded. Dabba mein fit ho (bounded) + dhakkan lagaa ho (closed). Examples: \( [0, 1] \) compact hai ✅ (closed + bounded). \( (0, 1) \) compact nahi ❌ (bounded hai lekin closed nahi). \( [0, \infty) \) compact nahi ❌ (closed hai lekin bounded nahi). \( \{1, 2, 3\} \) compact hai ✅ (finite set = closed + bounded).

\( x \) is a limit point of \( A \) if every neighborhood of \( x \) contains a point of \( A \) other than \( x \) itself. Note: \( x \) does NOT need to be in \( A \)!

Hinglish: Limit point = ek aisa point jiski har neighborhood mein set A ke elements ghuse rehte hain (khud ke alawa). Jaise ek celebrity ke ghar ke bahar hamesha fans khade rehte hain — celebrity ka ghar ek limit point hai fans ke set ka! Example: \( A = \{1/n : n \in \mathbb{N}\} = \{1, 1/2, 1/3, \dots\} \). Point 0 ek limit point hai \( A \) ka (bhale hi \( 0 \notin A \)!) kyunki 0 ke aas-paas hamesha set ke elements mil jaayenge.

A point \( x \in A \) is isolated if there exists some \( \varepsilon > 0 \) such that \( (x - \varepsilon, x + \varepsilon) \cap A = \{x\} \). It sits alone — no other set elements nearby.

Hinglish: Isolated point = set mein akela baitha hai, doosre elements se door. Example: Set \( A = \{0, 1, 2\} \) mein teeno points isolated hain kyunki har point ke aas-paas ek chhota sa gap hai jahan koi aur element nahi.

The closure of a set \( A \) is \( \overline{A} = A \cup A' \), where \( A' \) is the set of all limit points of \( A \). It is the smallest closed set containing \( A \).

Hinglish: Closure = original set + uske saare limit points ko andar le lo. Example: \( A = (0, 1) \) → \( \overline{A} = [0, 1] \) (endpoints 0 aur 1 limit points hain, unhe include kar diya). \( A = \{1/n\} \) → \( \overline{A} = \{0\} \cup \{1/n\} \) (0 ko include kiya).

\( \lim_{x \to c} f(x) = L \) means: for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( 0 < |x - c| < \delta \implies |f(x) - L| < \varepsilon \). Note: we don't care about \( f(c) \) itself, only what happens near \( c \).

Hinglish: Function limit = jab \( x \) kisi point \( c \) ke paas jaata hai (bina chhue), toh output \( f(x) \) kahan pahunchta hai. Example: \( \lim_{x \to 2} x^2 = 4 \). Jaise jaise \( x \) 2 ke paas jaata hai (1.9, 1.99, 1.999...), \( x^2 \) 4 ke paas jaata hai (3.61, 3.96, 3.996...).

\( f \) is continuous at \( c \) if \( \lim_{x \to c} f(x) = f(c) \). Equivalently (\( \varepsilon-\delta \)): for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( |x - c| < \delta \implies |f(x) - f(c)| < \varepsilon \). \( f \) is continuous on a set \( A \) if it is continuous at every point of \( A \).

Hinglish: Continuous = function ka graph bina uthaye (bina pen uthaye) draw kar sako. Koi jump nahi, koi hole nahi. Example: \( f(x) = x^2 \) har jagah continuous hai ✅ (smooth curve). \( f(x) = \lfloor x \rfloor \) (floor function) integers pe discontinuous hai ❌ (har integer pe step/jump hai).

\( f \) is uniformly continuous on \( A \) if for every \( \varepsilon > 0 \), there exists a single \( \delta > 0 \) (that works for all points simultaneously) such that \( |x - y| < \delta \implies |f(x) - f(y)| < \varepsilon \) for all \( x, y \in A \).

Hinglish: Regular continuity mein \( \delta \) har point \( c \) pe alag ho sakta hai. Uniform continuity mein ek hi \( \delta \) sabke liye kaam karta hai! Example: \( f(x) = x^2 \) poore \( \mathbb{R} \) pe uniformly continuous nahi hai (bade x pe graph bahut steep ho jaata hai, chhota \( \delta \) chahiye). Lekin \( [0, 10] \) pe uniformly continuous hai (bounded interval pe continuous ⟹ uniformly continuous by theorem!).

The derivative of \( f \) at \( c \) is \( f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h} \), if this limit exists. It represents the instantaneous rate of change — the slope of the tangent line at \( c \).

Hinglish: Derivative = function kitni tezi se badh/ghut raha hai us point pe. Slope of tangent line. Example: \( f(x) = x^2 \), \( f'(x) = 2x \). Point \( x = 3 \) pe slope = \( 2 \times 3 = 6 \). Matlab function yahan tezi se badh raha hai. Point \( x = 0 \) pe slope = 0 (graph flat hai yahan — minimum point!).

A secant line passes through two points on a curve. Its slope is \( \frac{f(x) - f(c)}{x - c} \) (average rate of change). A tangent line touches the curve at one point — it is the limit of secant lines as the two points merge: \( \text{slope} = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \).

Hinglish: Secant = curve ke 2 points ko jodne wali line (jaise do pehaadon ke beech ki rassee). Tangent = curve ko sirf 1 point pe chhune wali line (jaise tire ko road chhoo raha hai). Jab secant ke dono points ek saath mein mila dete hain, secant tangent ban jaati hai — aur uska slope = derivative!

Assume the opposite of what you want to prove. Show that this assumption leads to a logical impossibility (contradiction). Therefore, the original statement must be true.

Hinglish: "Ulta sochke dikhao ki galat hai." Maan lo tumhe prove karna hai "A sach hai". Pehle maano "A jhooth hai". Phir dikhao ki isse kuch aisa nikalta hai jo possible hi nahi. Matlab A jhooth ho hi nahi sakta → A sach hai! Example: √2 irrational hai — hum maante hain ki rational hai (p/q form mein), phir dikhate hain ki p aur q dono even aa jaate hain, jo contradiction hai (kyunki humne maana tha ki p/q simplified form mein hai).

Step 1 (Base): Prove the statement for \( n = 1 \). Step 2 (Inductive): Assume true for \( n = k \), prove for \( n = k+1 \). Then the statement is true for all \( n \in \mathbb{N} \).

Hinglish: Domino effect! Pehla domino gira do (base case). Phir dikhao ki agar koi bhi \( k \)-wala domino gira, toh \( k+1 \)-wala bhi girna guaranteed hai (inductive step). Isse saare domino gir jaayenge!

For any \( a, b \in \mathbb{R} \): \( |a + b| \leq |a| + |b| \). This is one of the most-used tools in analysis proofs.

Hinglish: Triangle ki kisi bhi ek side ki length baaki dono sides ke sum se chhoti ya equal hoti hai. Numbers ke liye: do numbers ka sum ka absolute value ≤ unke individual absolute values ka sum. Example: \( |3 + (-5)| = |-2| = 2 \leq |3| + |-5| = 8 \). Ye inequality ε-δ proofs mein baar baar lagti hai!

\( \mathbb{N} = \{1, 2, 3, \dots\} \) — the counting numbers. Born from the need to count things.

\( \mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\} \) — \( \mathbb{N} \) plus \( 0 \) and negatives. Now subtraction always works (\( 3 - 5 = -2 \in \mathbb{Z} \)).

\( \mathbb{Q} = \left\{ \frac{a}{b} : a,b \in \mathbb{Z}, b \neq 0 \right\} \) — every ratio of integers. \( \mathbb{Q} \) is a field: now division (except by 0) always works.

\( \mathbb{R} = \mathbb{Q} \cup \text{Irrationals} \) — every gap filled. \( \mathbb{R} \) has the Least Upper Bound (completeness) property.

A real number is irrational if it cannot be expressed as \( \frac{a}{b} \) where \( a, b \in \mathbb{Z} \) and \( b \neq 0 \). Their decimal expansions are non-terminating and non-repeating.

Hinglish: Irrational = jo \( p/q \) form mein likha hi nahi ja sakta. Inka decimal kabhi khatam nahi hota aur kabhi repeat bhi nahi karta. Famous Examples:

• \( \sqrt{2} = 1.41421356\dots \) — sabse pehla discovered irrational number
• \( \pi = 3.14159265\dots \) — circle ka circumference/diameter
• \( e = 2.71828182\dots \) — Euler's number (growth ka formula)
• \( \sqrt{3}, \sqrt{5}, \ln 2 \) — aur bhi bahut hain!

\( \mathbb{R} \) is complete: every non-empty subset of \( \mathbb{R} \) that is bounded above has a least upper bound (supremum) in \( \mathbb{R} \). This is what makes \( \mathbb{R} \) fundamentally different from \( \mathbb{Q} \).

Hinglish: \( \mathbb{Q} \) mein "chhede" (holes) hain — jaise \( \{x \in \mathbb{Q} : x^2 < 2\} \) ka sup = \( \sqrt{2} \), jo \( \mathbb{Q} \) mein hai hi nahi! \( \mathbb{R} \) mein saare chhede bhar diye gaye hain — har bounded set ka sup exist karta hai \( \mathbb{R} \) ke andar. Yahi poore course ki buniyaad hai!