Discrete mathematics :

8. Suppose that Rand S are reflexive relations on a set A with |A| = n > 0. State whether following statements are true or false? (1) (points will be given only if all answers are correct) (a) R S is reflexive (b) RS is irreflexive (c) So R is reflexive 9. Let A be a set with |A| = n, R be a relation on A. Prove that R=RU R U...UR (2) 10. Let f A B and g: B C be two functions. If g of is onto, is g onto? What about f? (1) 11. If (A,) and (B,) are two partial orders. Prove that the relation (A x B,) is also a partial order with (a, b) (c,d) iff a c in A and b d in B. (2) 12. Let A= {1,2,3,4,5,6} compute (5, 6, 3) o (4, 1, 3, 5) (1) 13. Define satisfiability and give an example proposition along with truth value assign- ment (2) 14. A relation R is defined on Z+ as a R biff a+b is even. Is R transitive? Give reasons (1) 15. Let A={1,2,5,8} and let R ={(a,b) | a divides b}. How many distinct topological sorts are there on this partial order relation ? Write all of them (1) 16. Let A= Z+ x Z+ and a relation R is defined on A as follows. (a, b) R (a', b') if ala' and bb. Is R a poset? If so prove otherwise disprove (3) 17. The matrix of a relation R is given to you. Outline a procedure to find if R is a partial order? (2)