Posted by **vilma** on Friday, October 26, 2012 at 6:15pm.

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+.