#### rootmeister64

##### New member

- Joined
- Nov 7, 2020

- Messages
- 10

Given: M = {1, 2,…, m} for m ∈ ℕ and N = {0, 1,…, 9}

f: M ⟼ N, f (x) = 2⋅x (mod 10)

**a) Give the Inverse image of f when m = 3.**

M = {1, 2, 3}

f(1) = 2, f(2) = 4, f(3) = 6

Therefore the image is: f(x)-1={2, 4, 6}(Is this a correct notation?)

**b) Assumption: g is a bijective mapping from M to N. How many maps h: M ⟼ N are there then?**

Since there is no limit for m, I would say that there is an infinite amount of mappings from M to N.

**c) How many permutations of N with 9 cycles are there?**

Since we have 10 elements in N there will always be a cycle with 2 elements.

Only the cycle with two elements will ‘change’ the permutation. Therefore we have 10 * 9 = 90 cycles. However since (a, b) = (b, a) we have to divide this amount by 2. Thus there are only 45 permutations.

**d) Give all 3-multisets of {0, 1}.**

{0, 0, 0}, {0, 0, 1}. {0, 1, 1}, {1, 1, 1}

**e) How many injective mappings are there from {0, 1} into the image of f, for m ≥ 5?**

The image of f for m ≥ 5 is {0, 2, 4, 6, 8}. Thus we have to map from {0, 1} to {0, 2, 4, 6, 8}.

For 0 we have 5 options and for 1 we have 4 options. Therefore 5*4 = 20 possible mappings.