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 ξ

b. Let A denote event that 1 wins the tournament. Then List outcomes in A.

c. Let B denote the event that 2 gets into the championship game. Then List outcomes in B.

d. What are the outcomes in A∪B and in A∩B? What are the outcomes in A'?

Answer:ξ = {1324, 1342, 1423, 1432, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 4123, 4132, 4213, 4231}

A = {1324, 1342, 1423, 1432}

B = {2314, 2341, 2413, 2431, 3214, 3241, 4213, 4231}

AuB = {1324, 1342, 1423, 1432, 2314, 2341, 2413, 2431, 3214, 3241, 4213, 4231}

AnB = ∅

A' = {2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 4123, 4132, 4213, 4231}

Step-by-step explanation:

Given that 1 will play 2 and 3 will play 4 in the first round.

If 1324 means I beats 2, 3 beats 4 and in the championship finals, 1beats 3 and 2 beats 4

Then possible outcomes = ξ = {1324, 1342, 1423, 1432, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 4123, 4132, 4213, 4231}

If A denotes the outcome that "1" wins the event, then

A = {1324, 1342, 1423, 1432}

If B denotes the event that "2" gets into the championship game final then,

B = {2314, 2341, 2413, 2431, 3214, 3241, 4213, 4231}

AuB is the union of elements in A and B and this gives:

AuB = {1324, 1342, 1423, 1432, 2314, 2341, 2413, 2431, 3214, 3241, 4213, 4231}

AnB is the common elements in A and B which is an empty set, this is denoted as:

AnB = ∅

A' means the elements that are not in A but are in the sample space.

A' = {2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 4123, 4132, 4213, 4231}