Question

Determine whether each of the following statements is true or false. Justify
your answer.
(a) x 2 fxg
(b) fxg  fxg
(c) fxg 2 fxg
(d) fxg 2 ffxgg
(e) ;  fxg
(f) ; 2 fxg
2. 4 pts. Consider the experiment of rolling two distinguishable fair dice. Let A be the
event where the two numbers appearing are relatively prime to each other. List the
elements of A.
3. 2 pts. A hand of ve cards is drawn from a standard deck of 52 cards. Let B be the
event that the hand contains ve aces. List the elements of B.
4. 8 pts. Let U = fa; b; c; d; e; f; gg;W = fa; b; dg;X = fb; d; eg, and Y = fa; e; f; g; g.
Find the following:
(a) W [ Y
(b) (X \W) [ Y
(c) Y \ (W [ X)
(d) (W \ Y )n(Y [ X)
5. 14 pts. Express each of the following events in terms of the events A;B; and C as
well as the operations of complementation, union and intersection. In each case, draw
the corresponding Venn-Euler diagram.
(a) at least one of the events A;B;C occurs;
(b) at most one of the events A;B;C occurs;
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(c) none of the events A;B;C occurs;
(d) all three events A;B;C occur;
(e) exactly one of the events A;B;C occurs;
(f) events A and B occur, but not C;
(g) either A occurs or, if not, then B also does not occur.
6. 5 pts. Denote the complement of any set X as Xc. Show that for any two sets A
and B, (A [ B)c = Ac \ Bc.
7. 5 pts. Prove that n3 + 5 is divisible by 4 for all n 2 Z+.