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+.
(a) To determine whether the statement "x < 2" is true or false, we need more context. Is "x" a variable that represents a number? In that case, we need to compare it to a specific value or provide additional information. Without further information, we cannot determine the truth value of this statement.
(b) To determine whether the statement "fxg � fxg" is true or false, we need to understand the meaning of "fxg". It seems to represent a set or a function, but we require more context to determine its properties. Without further information, we cannot determine the truth value of this statement.
(c) To determine whether the statement "fxg 2 fxg" is true or false, we need to understand the meaning of "2" in this context. It might represent the subset relation, meaning "fxg" is a subset of "fxg". In that case, the statement would be true because any set is a subset of itself.
(d) To determine whether the statement "fxg 2 ffxgg" is true or false, we need to understand the meaning of "2" in this context. It might represent the proper subset relation, meaning "fxg" is a proper subset of "ffxgg". Without further information, we cannot determine the truth value of this statement.
(e) To determine whether the statement "; � fxg" is true or false, we need to understand the meaning of ";" and "fxg". Without further information, we cannot determine the truth value of this statement.
(f) To determine whether the statement "; 2 fxg" is true or false, we need to understand the meaning of ";" and "fxg". Without further information, we cannot determine the truth value of this statement.
2. The event A represents rolling two numbers on two distinguishable fair dice that are relatively prime to each other. To determine the elements of A, we need to list all possible outcomes where the two numbers are relatively prime. For example, (1,2), (1,3), (1,4), (1,5), (1,6), (2,3), (2,5), (3,4), (3,5), (4,5), and (5,6) are all elements of A.
3. The event B represents drawing a hand of five aces from a standard deck of 52 cards. Since there are only four aces in a deck, it is impossible to draw five aces in one hand. Therefore, the event B has no elements.
4.
(a) W [ Y represents the set of elements that belong to either W or Y (or both). By combining the elements of W and Y, we have W [ Y = {a, b, d, e, f, g}.
(b) (X \ W) [ Y represents the set of elements that belong to either (X \ W) or Y (or both). By subtracting the elements of W from X, we have X \ W = {b, d, e}. Combining this with Y, we have (X \ W) [ Y = {a, b, d, e, f, g}.
(c) Y \ (W [ X) represents the set of elements in Y that do not belong to the set (W [ X). By combining the elements of W and X, we have W [ X = {a, b, d, e}. Taking the elements that are not in this set from Y, we have Y \ (W [ X) = {f}.
(d) (W \ Y) n (Y [ X) represents the set of elements that belong to both (W \ Y) and (Y [ X). By subtracting the elements of Y from W, we have W \ Y = {b, d}. The set (Y [ X) is equal to Y since X is a subset of Y. Therefore, (W \ Y) n (Y [ X) = {f}.
5. (a) The event "at least one of the events A, B, C occurs" can be expressed as (A U B U C).
(b) The event "at most one of the events A, B, C occurs" can be expressed as (A U B U C)'.
(c) The event "none of the events A, B, C occurs" can be expressed as (A' n B' n C').
(d) The event "all three events A, B, C occur" can be expressed as (A n B n C).
(e) The event "exactly one of the events A, B, C occurs" can be expressed as (A' n B n C) U (A n B' n C) U (A n B n C').
(f) The event "events A and B occur, but not C" can be expressed as (A n B n C').
(g) The event "either A occurs or, if not, then B also does not occur" can be expressed as (A U B').
6. To prove that (A U B)' = A' n B', we need to show that an element is in the complement of (A U B) if and only if it is in both A' and B'.
Assume x is an element in (A U B)'. This means x does not belong to A U B. Therefore, x does not belong to both A and B. Hence, x must belong to both A' and B', as it is not in A or B.
Now, assume x is an element in both A' and B'. This means x is not in A or B. Therefore, x does not belong to A U B, making it an element in (A U B)'.
Since every element in (A U B)' is also in A' and B', and vice versa, we can conclude that (A U B)' = A' n B'.
7. To prove that n^3 + 5 is divisible by 4 for all n in Z+ (positive integers), we need to show that (n^3 + 5) is always divisible by 4.
By inspecting different values of n, we can observe that n^3 + 5 is always divisible by 4.
For n = 1, (1^3 + 5) = 6, which is divisible by 4 (1.5 times).
For n = 2, (2^3 + 5) = 13, which is not divisible by 4.
For n = 3, (3^3 + 5) = 32, which is divisible by 4 (8 times).
By continuing this pattern, we can see that for every n in Z+, n^3 + 5 is divisible by 4. Thus, we have proven that n^3 + 5 is divisible by 4 for all n in Z+.