Question

Four universities-1, 2, 3, and 4-are participating in a holiday basketball tournament. In the first round, 1 will play 2 and 3 will play 4. Then the two winners will play for the championship, and the two losers will also play. One possible outcome can be denoted by 1324 (1 beats 2 and 3 beats 4 in first-round games, and then 1 beats 3 and 2 beats 4). (Enter your answers in set notation. Enter EMPTY or for the empty set.) (a) List all outcomes in (b) Let A denote the event that 1 wins the tournament. List outcomes in A (c) Let B denote the event that 2 gets into the championship game. List outcomes in B. (d) What are the outcomes in A UB? AUB- What are the outcomes in A n B? What are the outcomes in A'?

Answer 1

a) Possible outcomes are

1 beats 2, 3 beats 4 then 1 beats 3, 2 beats 4 -- 1324

1 beats 2, 3 beats 4 then 1 beats 3, 4 beats 2 -- 1342

1 beats 2, 4 beats 3 then 1 beats 4, 2 beats 3 -- 1423

1 beats 2, 4 beats 3 then 1 beats 4, 3 beats 2 -- 1432

.........................................................................................

........................................................................................

The set is

$S=\left \{ 1324,1342, 1423, 1432, 2314,2341, 2421, 2431, 3124,3142,3214,3241, 4123,4132,4213,4231 \right \}$

b) When 1 wins the tournament, set (1 appears as first in each outcome)

$A=\left \{ 1324,1342, 1423, 1432 \right \}$

c) The outcomes of such that 2 gets into the championship game is (2 appears as first or second in each outcome)

$B=\left \{ 2314,2341, 2421, 2431,3214,3241, 4213,4231 \right \}$

d) The set

$A\cup B=\left \{ 1324,1342, 1423, 1432, 2314,2341, 2421, 2431,3214,3241, 4213,4231 \right \}$ e) $A\cap B=\phi$ . are disjoint.

f) Complement of   is

$A'=\left \{ 2314,2341, 2421, 2431, 3124,3142,3214,3241, 4123,4132,4213,4231 \right \}$