Name: SSN: Grade:MA334 EXAM 3 December 1997I pledge my honor that I have abided by the Stevens Honor System.1. (15pts) Prove by induction: For all integers n ≥ 1,n2+5n + 6 is even.2 (15pts) Prove by induction: For all integers n ≥ 2, n can be written as a product of primes.3 (15pts)• Find an equivalence relation R on the set A = {1, 2, 3, 4, 5,x}.• Find the distinct equivalence classes of your relation R.4 (10pts)Given the relation R below, draw (in a separate diagram) the transitive closure Rt.5 (20pts) Consider the “divides” relation on the set A = {2, 3, 5, 6, 15, 20, 24, 120}, i.e., xRy if andonly if y = xz for some integer z.• Draw the Hasse diagram H.(6)• Find all maximal elements. (3)• Find all minimal elements. (3)• Find all greatest elements. (2)• Find all least elements. (2)• Find, if possible, a 3 element antichain in R containing the element 24. (4)6 (15pts) Consider the relation R1on A and the relation R2from A to B given below. Find• R1R2• R217 (10pts) Let T be an arbitrary transitive relation on a set A. Prove that the inverse T−1is