# Discrete Math

**Question description**

I need 12 problems done for a discrete math class.....the two below are examples....there are over 30 problems to choose from (you can pick)

Example 1

**12.
a) **Prove that *(*cos *θ *+ *i *sin *θ)*2 _ cos 2*θ *+ *i *sin 2*θ*,where *i *∈
**C **and *i*2 _ −1.

**b)
**Using induction, prove that for
all *n *∈ **Z**+,*(*cos *θ *+ *i *sin *θ)**n *_ cos *nθ *+ *i *sin *nθ.*

(This result is known as *DeMoivre’s Theorem*.)

**c)
**Verify that 1 + *i *_√2*(*cos 45◦ + *i *sin 45◦*)*, and compute*(*1 +
*i)*100.

Example 2

2. For each of the following functions f : Z→Z, determine

whether the function is one-to-one and whether it is onto. If the

function is not onto, determine the range f (Z).

a) f (x) x + 7 b) f (x) 2x − 3

c) f (x) −x + 5 d) f (x) x2

e) f (x) x2 + x f ) f (x) x3

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