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 2θ + i sin 2θ,where i ∈
C and i2 _ −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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