3. On the relation R on the set {1, 2, 3, 4, 5} defined by the rule (x, y) e R if x+y<6. a) List the elements of R. b) List the elements of R'. c) Find the domain of R. d) Find the range of R. e) Find the domain of R'. f) Find the range of R'. g) Is the relation R an equivalence relation? If so, list the equivalence classes.

Determine whether the given relation is an equivalent relation on (1, 2, 3, 4, 5} If the relation is an equivalence relation, list the equivalence classes (x, y E {1,2, 3, 4, 5} . ) 3, 4, 5}. 1. {(1,1), (2, 2) , (3, 3) , (4, 4) , (5, 5) , (1, 3) , (3, 1)} 2. {(x, y) | 3 divides x + y}

Let R be the relation on the set of all mathematicians that contains the ordered pair (a, b) if and only if a and b have written a published mathematical paper together. a) Describe the relation R[2] in the context of the problem.b) Describe the relation R* in the context of the problem.c) The Erdos number of a mathematician is 1 if this mathematician wrote a paper with the prolific Hungarian mathematician Paul Erdos. The Erdos number is 2 if a mathematician didn’t write a joint paper with Erdos but wrote a joint paper with someone who wrote a joint paper with Erdos, and so on (except that the Erdos number of Erdos himself is 0). Give a definition of the Erdos number is terms of paths in R. List a name of a mathematician with number: 1, 2, and 3

prove why each relation has or does not have theproperties: reflexive, symmetric, anti-symmetric, transitive. 1) Let A = { set of all people }, relation R: A x A where R = { (a,b) | a,b ∈ A, a is at least as tall as b }

prove why each relation has or does not have theproperties: reflexive, symmetric, anti-symmetric, transitive. 4) Z+ is the set of positive integers, relation R: Z+x Z+, a,b,c,d ∈ Z+; (a,b),(c,d) ∈ R if an only if   a + d = b + c.