hellohomeworkassignment2.pdfHomework Assignment 2

Answer each of the questions below and submit your answers as a PDF document through

Canvas by the due date.

1. Suppose that you have the Boolean function f ( x , y )=¬(x∧ y)∨ y . Simplify this

expression.

2. Suppose that you had a set of 25 items. How many different subsets with exactly 7

items are there? How many different subsets with less than 3 items are there?

3. Suppose that you have 9 people in a line waiting for ice cream. In how many different

orders could the seven people be served?

4. Suppose that a bowl has six blue balls, five red balls, and four green balls.

a. What is the probability of selecting a green ball?

b. What is the probability of selecting a green or red ball?

c. What is the probability of selecting a green ball, then a red ball, assuming

that you do not put the balls back in the bowl?

5. Let S and T be outcomes. The probability P(S) is .3, the probability P(T) is .5, the

probability P(S∩T) is .35. How do you know that outcomes S and T are not

independent? What is the conditional probability P(S|T)? What is the conditional

probability P(T|S)?

6. Suppose that S and T are two mutually exclusive outcomes. What is P(S|T)? What is

P(T|S)? Give a numeric answer, not symobolic. How do you know this numeric result?

7. Suppose that S and T are independent outcomes and that P(S) = .4. What is P(S|T)

as a numeric result? How do you know this?

8. Prove that a complete binary tree of height h has 2h+1−1 nodes.

9. Prove that a complete binary tree with n nodes has height lg⌈n+ 1⌉−1 .

10. Prove that while f (n)=2 n2 +n+30∈O( g( n)) , where g(n)=n2 , f (n)∉ o( g(n)) . (Hint,

use the limit approach).

11. Another definition for little-o is that

1) f (n)∈o(g(n)) if ∀ c >0, ∃n0 such that f (n)≤cg(n), ∀ n>n0 .

The constant n0 may depend on the choice of c. In the definition of big-O, there only

need be at least one constant c for which the inequality holds. Show that n n∉o (n) ,

but that n∈O(n) . (In the first case, find a constant c for which 1) does not hold).

12. Consider the function f (n)=lg lg n=lg(lg ( n)) . Assume the each lg is log base 2. For

what value of n will lglg produce a value greater than 10? Where does this function

fall in the big-O ordering of functions presented in the textbook?

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