Advanced Mathematics for Computer Science HNCST409 KTU BTech Honors 2024 Scheme- Dr Binu V P

About Me Syllabus Advanced Mathematics for Computer Science HNCST409 BTech Honors 2024 Model Question Paper and Answers Module- I Number Theory  Introduction Group , Ring and Fields Divisibility Modular Arithmetic and Congruences Equivalence Relation - Congruence Linear Congruences Solving simultaneous congruences- CRT Modular Inverses      Euclidean algorithm      Extended Euclidean algorithm  Euler’s and Fermat’s little theorem  Euler's Totient Function Prime Numbers and Prime-Power Factorization Fermat and Mersenne Prime Primality Testing and Factorization Wilson's Theorem Pseudo Primes and Carmichael Numbers Primality testing       Miller-Rabin      AKS Galois Field (introduction and basic operations) RSA cryptosystem: key generation - encryption/decryption  Hash functions (introduction) Module - II Optimization Convex and Non Convex sets Convex Hull Identifying convex sets Applicatio...

  What Is an Equivalence Relation? Let A A  be a set. A relation \sim ∼ on A A  is called an equivalence relation if it satisfies three properties : 1. Reflexive For every a ∈ A a \in A , a ∼ a a \sim a 2. Symmetric For all a , b ∈ A a, b \in A , a ∼ b ⇒ b ∼ a a \sim b \Rightarrow b \sim a 3. Transitive For all a , b , c ∈ A a, b, c \in A , a ∼ b  and  b ∼ c ⇒ a ∼ c a \sim b \text{ and } b \sim c \Rightarrow a \sim c Why Equivalence Relations Are Important Equivalence relations allow us to: Group elements into equivalence classes Treat different objects as “essentially the same” Build structures like modular arithmetic , finite fields , and quotient sets Congruence Modulo n n n Let n ∈ Z , n > 0 n \in \mathbb{Z}, n > 0 . For integers a a  and b b , we say: a ≡ b ( m o d n ) a \equiv b \pmod{n} if and only if: n ∣ ( a − b ) n \mid (a - b) That is, a − b a - b  is divisible by n n . Claim Congruence modulo n n n is...