2/2. prove that if G is Euler then so is L (G).
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- 6. Prove that |xy| = |x|ly| for all real numbers x, y.The Euler totient function is defined as ϕ (m) = |{k : 1 ≤ k ≤ m gcd(k, m) = 1}|, or rather the number of relatively prime positive integers smaller than or equal to m. The Task is to conclude if : X = z1p1,z2p2 , ........., zkPk THEN, ϕ(x) = x (1 - 1/p1), (1 - 1/p2),....., (1 - 1/P)5. Prove that of has a zero of order m at zo if and only if f(20), f'(zo), . . ., f(m−1) (20) are all equal to zero, but f(m) (20) 0.
- 7.1 245.6.15. Let f: A → B be an arbitrary function. (a) Prove that if f is a bijection (and hence invertible), then f-'(f(x)) = x for all x E A, and f(ƒ¯'(x))= x for all x E B. (b) Conversely, show that if there is a function g: B → A, satisfying g(f(x)) = x for all x € A, and f(g(x)) = x for all x E B, then f is a bijection, and f-1 = g.A E Z₂ x Z₂ ({0},+) ZA Z₁ x Zo Which one of the following is isomorphic to Z4 x Z5/((1,2))?
