DISCRETE MATHEMATIC 1 How many integers from 0 through 30 in

DISCRETE MATHEMATIC

1) How many integers from 0 through 30, including 0 and 30,
must you pick to be sure of getting at least one integer
a) that is odd?
b) that is even?
Explain your answer.

2) 700 people are in a group. We\'ll use F to represent the first
initial of their first name, L for the first initial of their last name.
Each person is identified with a pair (F,L).
At least how many people from this group have the same pairs
(F,L) of initials of their first and last names?

3) Find the big O estimate for functions f(x), xR
a) f(x) = 4x3 + x2logx;
b) f(x) = (x5+ 3x2 +2)/(x3 + 6);

4) Find the smallest integer n so that function f(x) is O(xn),xR for
a) f(x) = 3x5 + (logx)4;
b) f(x) = (x2 + 1)(2x4 + 3x - 8);

5) Show that 3x2 + log2x2 is Theta(x2).
log2x is log base 2 of x.

Solution

1. There are 16 even integers and 15 odd integers between 0 and 30 inclusive.So at least one number of what we pick is even we must pick 16 integers. Anything less than 16 will not be sufficient because we could pick all the odd integers .And we get at least one odd number we must pick 17 integers. Any thing less than 17 is insufficient to guarantee an odd outcome because we could pick all the even integer.

2.Since there are 26 options for each of f and l, there are 26*26 = 676 unique pairs (f,l). Since there are 700 people in the group, we could have 700 - 676 + 1 = 25 people all with the same first and last name initials and 675 with unique initial pairs (f,l). So there are at least 25 people who share their initial pair with someone else in the group.