, the union of  and   is the set of all elements in  , , or both. Merge the two sets to get:

The complement of a set is the set of all elements in the universal set that are not in the set. The only elements in  not in  are 4 and 8, so 

 refers to the empty set, the set with no elements;  if and only if the two sets have no elements in common. In four of these cases,  and  share an element, which in each of these four choices is underlined:

 and  do not have an element in common, so this is the right choice.

 is the set of all elements in both  and .

We can test each set and determine which elements are shared by that set and :

If :

then 

If :

then 

If :

then 

If :

then 

If :

then 

This is the correct choice.

For the roll to be unfavorable to the event that at least one of the dice is 5 or 6, both dice would have to be 1, 2, 3, or 4. There are  ways out of 36 that this can happen, so there are  ways for either or both of the two dice to be 5 or 6. Since half of 36 is 18, the probability of this event is greater than .

The product of the two dice will be an odd number only if both dice are odd; the rolls favorable to that event are:

,

or nine out of the thirty-six possible rolls. This makes twenty-seven of the thirty-six equally probable rolls favorable to getting an even product. Since half of 36 is 18, the probability of getting an even product is greater than .

This makes (a) the greater quantity

The easiest way to answer this is to try to match each element in the first set to one the second set as follows:

...

In other words, each element in the set in (a) is paired with the element in the set in (b) that is its double. Since there is a one-to-one correspondence, the two sets are of equal aize, and (a) and (b) are equal quantities.

The easiest way to answer this is to try to match each element in the first set to one in the second set as follows:

...

Since there is a one-to-one correspondence between the elements of the two sets, (a) and (b) are equal.

Out of a possible thirty-six rolls, the following result in a product of  or greater:

This is ten equally probable rolls out of thirty-six, resulting in a probability of 

.

Since , (b) is the greater quantity.

This question can be most easily answered by matching each element in the set in (a) with the next consecutive integer, which is in the set in (b):

...

Every element in the second set has a match, but there is an unmatched element in the first set. Therefore (a) is the greater quantity.

The quantities are equal. This can be proved as follows:

The sum of the integers from  to  is

.

(b) is twice this:

This is the same value as (a), the sum of the even integers from  to .