Consider the set of positive even integers S = {2, 4, . . . , 3000}. i. What is the cardinality of this set? ii. How many of these integers are divisible by 3? iii. How many of these integers are...

1. We are given the following set of positive even integers:

 2,4,6, ,3000 .S 

i. We wish to determine the cardinality of S.

The cardinality of S is the number of even positive integers up to 3000, which is equal
to 3000/2, or 1500.
ii. We wish to determine how many of these integers are divisible by 3.

Since 2 and 3 are relatively prime, the elements of S that are divisible by 3 are the
positive integers up to 3000 that are divisible by 2 3 6.  The number of these is
3000/6, or 500.
iii. We wish to determine how many of these integers are divisible by 5.

Since 2 and 5 are relatively prime, the elements of S that are divisible by 5 are the
positive integers up to 3000 that are divisible by 2 5 10.  The number of these is
3000/10, or 300.
iv. We wish to determine how many of these integers are divisible by 3 and 5.

Since 2, 3, and 5 are pairwise relatively prime, the elements of S that are divisible by
3 and 5 are the positive integers up to 3000 that are divisible by 2 3 5 30.   The
number of these is 3000/30, or 100.
v. We wish to determine how many of these integers are divisible by 3 or 5.

The set of integers in S that are divisible by 3 or 5 is the union of the set 3S of
integers in S that are divisible by 3 and the set 5S of those that are divisible by 5.
However, these two sets have a nonempty intersection, namely 3 5 15 ,S S S  whose
cardinality we computed in the previous part. Thus we have
   3 5 3 5 3 5# # # #
500 300 100
700.
S S S S S S    
  

vi. We wish to determine the least number n of distinct integers that must be chosen from
S so that at least one of them is divisible by 3.
The number n is equal to one more than the number of elements of S that are not
divisible by 3. Thus we have
3# # 1
1500 500 1
1001.
n S S  
  

vii. We wish to determine the least number n of distinct integers that must be chosen from
S so that at least one of them is divisible by 5.
The number n is equal to one more than the number of elements of S that are not
divisible by 5. Thus we have
3# # 1
1500 300 1
1201.
n S S  
  

viii. We wish to determine the least number n of distinct integers that must be chosen from
S so that at least one of them is divisible by 3 or 5.
The number n is equal to one more than the number of elements of S that are not
divisible by 5. Thus we have
 3 5# # 1
1500 700 1
801.
n S S S   
  

2. We assume that in the American election on Nov. 6, 2012, there are two choices for each
of the 469 seats (one president, 33 senators, and 435 representatives), namely Democrat
or Republican.
i. We wish to determine the total number of possible outcomes.

Since there are two possible outcomes for each seat and 469 seats, the total number of...