X


Expected Value and Variance




Q. Let X be the number that comes up when a fair die is rolled. What is the expected value of X?



Q. A fair coin is flipped three times. Let S be the sample space of the eight possible outcomes, and let X be the random variable that assigns to an outcome the number of heads in this outcome. What is the expected value of X?



Q. What is the expected value of the sum of the numbers that appear when a pair of fair dice is rolled?




Relations and Their Properties






Example of binary relations :



Q. Let A ={0, 1, 2} and B ={a, b}. Then {(0, a), (0, b), (1, a), (2,b)} is a relation from A to B. This means, for instance, that 0 R a, but that 1 R b. Relations can be represented graphically, as shown in Figure below, using arrows to represent ordered pairs. Another way to represent this relation is to use a table, which is also done in Figure below.




Relations on a Set




Q. Let A be the set {1, 2, 3, 4}. Which ordered pairs are in the relation R ={(a, b) | a divides b}?



Q. Consider these relations on the set of integers: R 1 = {(a, b) | a ≤ b}, R 2 = {(a, b) | a > b}, R 3 = {(a, b) | a = b or a = −b}, R 4 = {(a, b) | a = b}, R 4 = {(a, b) | a = b + 1}, R 5 = {(a, b) | a + b ≤ 3}. Which of these relations contain each of the pairs (1, 1), (1, 2), (2, 1), (1, −1), and (2, 2)?





Q. How many relations are there on a set with n elements?




Properties of Relations




Q. Consider the following relations on {1, 2, 3, 4}: R1 ={(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}, R 2 ={(1, 1), (1, 2), (2, 1)}, R 3 ={(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}, R 4 ={(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}, R 5 ={(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}, R 6 ={(3, 4)}. Which of these relations are reflexive?





Q. Is the “divides” relation on the set of positive integers reflexive?



Q. How many reflexive relations are there on a set with n elements?



Q. R1 ={(a, b) | a ≤ b}, R2 ={(a, b) | a > b}, R3 ={(a, b) | a = b or a = −b}, R4 ={(a, b) | a = b}, R5 ={(a, b) | a = b + 1}, R6 ={(a, b) | a + b ≤ 3}. Which of these relations are symmetric and which are antisymmetric?





Q. Consider the following relations on {1, 2, 3, 4}: R1 ={(1, 1), (1, 2), (2, 1), (2, 2), (3, 4), (4, 1), (4, 4)}, , R 4 ={(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3)}. Which of these relations are transitive?



Q. Is the “divides” relation on the set of positive integers transitive?




Combining Relations






Q. Let A and B be the set of all students and the set of all courses at a school, respectively. Suppose that R1 consists of all ordered pairs (a, b), where "a" is a student who has taken course "b", and R2 consists of all ordered pairs (a, b), where "a" is a student who requires course "b" to graduate. What are the relations R1 ∩ R2, R1 ∪ R2, R1 ⊕ R2, R1 - R2 and R2 - R1 ?


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