Let R be a transitive relation on set A. Define Q to be {(r Prove or disprove that (RUR®)n (QU Qº) must be a refe

Theorem: Let R ⊆ A × A be a relation. Then R is transitive if and only if R ◦ R ⊆ R. Prove Theorem: show that R is transitive if and only if R ◦ R ⊆ R. No handwritten please

Let R and S be reflexive relations over set A. Prove or disprove that RnS is reflexive

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