Discrete mathematics

Find a transitive closure of the relation R on {a,b,c,d,e} given by R= {(a,b), (a,c), (a,e),(b,a), (b,c),(c,a), (c,b),(d,a,),(e,d)}

Discrete mathematics

(See attached file for full problem description) --- Let R1 and R2 be relations on a set A. represented by the matrices: M R1 0 1 0 M R2 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 find the matrices that represent ( show all work) a) R1 union R2 b) R1 intersection R2 c) R2 º R1 (composition) d) R1 º R1 (composition) e) R1 symmetric different R2 ---

4 Problems

(See attached file for full problem description) --- 1. Show that if A and B are countable and disjoint, then A B is countable. 2. Show that any set, A, of cardinality c contains a subset, B, that is denumerable. 3. Show that the irrational numbers have a cardinality c. 4. Show that if A is equivalent to B and C is equivalent to D, then A C is equivalent to B D. ---

2 Problems

6. Let (G, *) be a group. Show that each equation of either the form ax = b or the form xa = b has a unique solution in G. 7. Show that (R - {1}, *), where a * b = a + b + ab is a group

2 Problems

5. Let (A, *) be an algebraic structure, and suppose that A is associative, has an identity, e, and that has an inverse. Show that if ax = ay, then x = y. 8. Let G be a finite group with identity e, and let . Show that there is an with an = e (Hint: Consider the set {e, a, a2 , …, am }, where m is the number of elements of G and use cancellation. (problem 5))