2/2. prove that if G is Euler then so is L (G).

. (블블) k – 1 k+1 Let h : Z → Q × Q, given by h(k) : 2 2 Let G = {(m, m + 1) | m E Z}. Show that G C h(Z).

3. (.) Denote the number of derangements of {1, 2,..., n} by Dn. Using the formula for the number of the derangement that we found in class, provide a proof of the relation D, = (n – 1)(D-2 + Dn-1); (n = 3,4, 5, ...).

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)

Please write neatly and explain everything clearly

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.

5. Prove that f has a zero of order m at zo if and only if f(zo), ƒ'(zo),...‚ ƒ(m-¹) (zo) are all equal to zero, but f(m) (zo) ‡0.

. Let ƒ € C[0, 1] and f(0) = f(1). Prove: 3x0 = [0, 1], such that ƒ (xo) = ƒ (xo + ½}).

2.)

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