Mathematical Operations – Logical Reasoning
(MIT‑Level Study Notes for Science Olympiad – NSO)
Goal: Master the first‑principles view of every arithmetic and algebraic operation, understand their logical properties and edge cases, and be able to use them to prove theorems or solve Olympiad‑style problems.
1. Overview
| Operation | Symbol | Domain | Key Properties |
|---|---|---|---|
| Addition | (+) | (\mathbb{R}) | Commutative, associative, identity (0), inverse (-x) |
| Subtraction | (-) | (\mathbb{R}) | Defined as (a-b:=a+(-b)) |
| Multiplication | (\times) | (\mathbb{R}) | Commutative, associative, identity (1), inverse (1/x) ( (x\neq0) ) |
| Division | (/) | (\mathbb{R}\setminus{0}) | (a/b:=a\times(1/b)) |
| Exponentiation | (^) | (\mathbb{R}\times\mathbb{N}) | Power rules |
| Roots | (\sqrt[n]{,}) | (\mathbb{R}) | Inverse of exponentiation |
| Factorial | (n!) | (\mathbb{N}) | Product of first (n) positive integers |
| Binomial coefficient | (\binom{n}{k}) | (\mathbb{N}) | (\frac{n!}{k!(n-k)!}) |
| Modulo | (\bmod) | (\mathbb{Z}) | Equivalence classes |
Logical Reasoning: Operations are not only computational tools; they are logical connectives that transform statements. Mastery allows you to deduce implications, prove identities, and solve combinatorial arguments.
2. First Principles & Fundamental Properties
2.1 Addition
- Definition: For any (a,b\in\mathbb{R}), (a+b) is the unique number satisfying
[ a+b = c \iff a = c-b. ] - Axioms (Peano for integers; field axioms for reals):
- (a+0=a) (identity).
- (a+(-a)=0) (inverse).
- (a+b=b+a) (commutative).
- ((a+b)+c=a+(b+c)) (associative).
Proof sketch: Use induction on natural numbers then extend to integers via additive inverses, then to rationals/reals via density/field axioms.
2.2 Multiplication
- Defined analogously: (a\times b = c) iff (a = c / b) (when (b\neq0)).
- Axioms: identity (1), inverse (1/a) for (a\neq0), commutative and associative properties, distributive over addition: [ a(b+c)=ab+ac. ]
2.3 Exponentiation
- For (n\in\mathbb{N}), define (a^0=1) and (a^{n+1}=a^n\cdot a).
- Powers of non‑integer exponents: extend via logarithms: [ a^x = e^{x\ln a}, \quad a>0,,x\in\mathbb{R}. ]
2.4 Roots
- For (n\in\mathbb{N}), the (n)-th root (\sqrt[n]{a}) is the unique (b) such that (b^n=a) (when (a\ge0) or (n) odd).
- Relation: (\sqrt[n]{a} = a^{1/n}).
2.5 Factorials & Binomial Coefficients
- Factorial: (n! = \prod_{k=1}^{n} k).
- Combinatorial Interpretation: Number of ways to order (n) distinct items.
- Binomial: (\binom{n}{k}) counts (k)-subsets of an (n)-set.
- Pascal’s identity: (\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}).
2.6 Modular Arithmetic
- Definition: For integers (a,b,m>0), (a \equiv b \pmod{m}) iff (m\mid (a-b)).
- Operations mod (m) preserve congruence: [ a\equiv b,;c\equiv d \pmod{m};\Rightarrow; a\pm c\equiv b\pm d,; a\