A die is tossed twice and the number of dots facing up is counted and
noted in the order of occurrence.

a.   Find the sample space.
b.   Find the set A corresponding to the event "The number of dots
showing is even"
c.   Find the set B corresponding to the event "both dies are even"
d.   Does A imply B or does B imply A?
e.   Let C corrospond to the event "number of dots in die is differ by
1. Find A OR C , A AND C

Hi alpa101473-ga !!

a. Assuming that the question is asking about the values of each die,
the sample space is:

S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1,6), (2, 1), (2, 2),
(2, 3), (2, 4), (2, 5), (2,6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5),
(3,6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4,6), (5, 1), (5, 2),
(5, 3), (5, 4), (5, 5), (5,6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5),
(6,6)}


b. We must find pairs of tosses which sum 2, 4, 6, 8, 10 or 12; that
leaves:

A = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3),
(3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6,
4), (6, 6)}


c. If B is the event "both tosses have an even number of dots", then:
  
B = {(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4),
(6, 6)}


d. A is the event "the sum of the dots of both tosses is even"; and B
is the event "both tosses have an even number of dots", then if B
occurs the sum of the dots must be even and in consequence A has
happened. The conclusion here is that B imply A. The opposite is
false, for example if we obtain two ones A has happened, but B has
not, then A NOT imply B.


e. If C is the event "number of dots of the first toss differs by 1 to
the number of dots of the second toss", then:

C = {(1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4),
(5, 6), (6, 5)}

What "A AND C" event means?
This means that both events A and C have happened, so an element of
this set must be in both sets A and C, that means in the intersection
of A and C. Then the set A AND C is the intersection of the sets A, C.
The sum of the dots in C are all odd, then:
A AND C = empty.

What "A OR C" means?
This means that at least one of the events must have happened, so an
element of this set can be in the set A or in the set C, that is the
same to say an element of "A OR C" must be in the set A union C.
Then:
A OR C = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3,
3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2),
(6, 4), (6, 6), (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4,
5), (5, 4), (5, 6), (6, 5)}


I hope this helps you. If you find something unclear in the answer,
please post a request for an answer clarification before rate it.

Best regards.
livioflores-ga