Discrete Mathematics Answers

4. If the roots of the cubic az3 + bz2 + cz + d = 0 form an arithmetic progression α − β,

α, α + β, prove that (2b

2 − 9ac)b + 27a

2d = 0.

Let A={2,3,6,12}and let R and S be the following relations on A. xRyiff 2 / x–y, xSyiff 3 / x–y. Compute the following R^---,R^S,R U S,S^-1

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. x~y in R if |x-y|<4

Draw the Hasse diagram for divisibility on the set {1,2,3,4,6,8,12}. Do the maximal, minimal elements exist? If so, what are they? What is the greatest element?

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. m~n in Z if m=n mod 6. 51. Which of them are equivalence relations?

(a) "less than" on the set N

(b) "has the same shape as" on the set of all triangles

Prove that the relation ‘’Superset of ’’ is a partial order relation on the power set of S.

Let R={(1,3),(4,2),(2,4),(2,3),(3,1) } be a relation on the A={1,2,3,4}. Find the transitive closure of R using Warshall’s algorithm.

For real number x and y, we write xRy⇔x-y+√2 is an irrational number. Is the relation

(a) Equivalence

(b) Partial order

Let R be the relation from A={2,3,4,5} to B={3,6,7,10} defined by ‘x divides y’,

(a) thenR^(-1) is equal to:

(b) Is R an equivalence relation?

Determine the following relation is an equivalence relation or not. If the relation is an equivalence relation, describe the partition given by it. xRy if x>yx,y∈the set of all real numbers