Question

Which of the following mappings are homomorphisms? Monomorphisms? Epimorphisms? Isomorphisms? a) G= (R-(0), .), H= (R+,-); φ: G→H is given by φ(x)=국 b) Ga(R" ,-); ф: G-* G is given by ф(x)-Vx c) G-group of polynomials p(x) with real coefficients, under addition of polynomials; φ: G-→(R, +) is given by φ[P(x)]-PC) d) G is as in (c); ф: G-G is given by ф[P(x)-p'(x), the derivative of P(x) e) G-the group of subsets of {1,2,3,4,5) under symmetric difference; A-(1,3,4), and p: G G is given by ф(8)-4aB, for every B

Answer 1

ANSWE R deosdy φ is surJective becaue it ho.ky value d is not injective becausc4c2)-( z, 22- uros getting ore num bea2

CCodomtua) getig same inoge so not injective cleasy thir map it suajectiue berause fo each sea oe will get a po lynoma/ Hence Ф і an epimorphism So Sungedtive