Please include your scratch files. You must explain your answers (even for True/False, multiple choice and/or fill in the blank.) 1. Consider all bit strings of length 12. How many of them contain exactly four 1's? 2. Consider all bit strings of length 12. How many of them contain at least four 1's? 3. The relation A on (1, 2, 3, ...) (Le., on Z*) where a R b means a | b (i.e., a divides b). This relation is reflexive. Note: a R b means a is related to b in the sense that the pair (a, b) is an element of R. (Circle: TRUE FALSE). 4. The relation R on (1, 2, 3, ...) (Le., on Z') where a R b means a | b (i.e., a divides b). This relation is antisymmetric. Note: a R b means a is related to b in the sense that the pair (a, b) is an element of R. (TRUE or FALSE). 0 1 5. The matrix of a relation R is n. -1 1 1 0 . Determine if the relation is 1 0 a) Reflexive Circle: Yes No b) Symmetric Circle: Yes No c) Antisymmetric Circle: Yes No d) Transitive Circle: Yes No 6. The matrix of a relation R is Me Sketch the digraph of the relation on a piece of paper, scan or photograph your sketch and upload it with your scratch file. 7. Draw the matrix that represents the given relation. Use elements in the order given to determine rows and columns of the matrix. Ron (1, 2, 3, 4) where a R b means la - b| $ 1. 8. Find the reflexive closure of the relation R on the set (1, 3, 5, 7) if A is given by R = {(1, 1), (1, 3), (1, 7), (3, 1), 3, 7), (5, 3), (5, 5), (7, 3), (7, 5)). Which, if any, pairs must be added to A to produce the reflexive closure? Write out the set of pairs that must be added to A to make it reflexive.9. Suppose A and 5 are relations on (a, b, c, d) where R = {(a, b), (a, d), (b, c], (c, c), (d, a)) and $ = ((a, c), (b, d), (d, a)). Compute R * 5, the composition of A and 5. NOTE: Relation S should be calculated first. 10. How many ways are there to select 6 students from a class of 25 to serve on a committee? 11. Consider the relation R on Z where a R b means a' = b'. a) Is the relation reflexive? Circle: Yes No b) Is the relation symmetric? Circle: Yes No c) Is the relation antisymmetric? Circle: Yes No d) is the relation transitive? Circle: Yes No Note: a R b means a is related to b in the sense that the pair (a, b) is an element of the relation. 12. Find the reflexive, symmetric, and antisymmetric closures of the relation R . ((1, 2), (1, 4), (2, 3), (3, 1), (4, 2) ) a) Reflexive closure b) Symmetric closure c) Antisymmetric closure 13. The number of relations on a set with a elements is 2". (Circle: TRUE FALSE). 14. Staff at a certain rural hospital consists of 2 male doctors, 3 female doctors, and 4 nurses. How many ways are there to: a) form a medical rescue group of 3 members? b) form a rescue group of 3 members that has no doctors? c) form a rescue group of 4 members that has one female doctor, one male doctor, and 2 nurses? 15. Represent the relation R on (1, 2, 3, 4) (ordered by listing) by a directed graph, a matrix, and circle its properties. R = {(1, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 1), (4, 4)). Digraph Matrix Circle properties of R: Reflexive Symmetric Antisymmetric Transitive16. For the following relation on a the set of all integers, circle its properties. xy20 Reflexive Symmetric Antisymmetric Transitive 17. Write down the following relations as a set of ordered pairs if A and $ are defined as follows: R = {(1, 1), (1, 2), (2, 3), (3, 3), (4, 3), (4, 4)). 5 = {(1, 2), (2, 3), (2, 4), (3, 2), (3, 3), (4, 1) ) a) ROS b) ROS 1 0 1 1 18. MAE 1 1 0 1 0 Find the matrices M. and M.BI