Relation
Q1.
Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ______.Q2.
Let G be an arbitrary group. Consider the following relations on G: R1: \forall a,b \in G, aR_1b if and only if \exists g \in G such that a=g^{-1}bg R2: \forall a,b \in G, aR_2b if and only if a=b^{-1} Which of the above is/are equivalence relation/relations?Q3.
The time complexity of computing the transitive closure of binary relation on a set of n elements is known to beQ4.
Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s)) \in R if and only if p-s=q-r. Which one of the following is true about R?Q5.
A relation R is defined on ordered pairs of integers as follows: (x,y)R(u,v) if x\ltu and y\gtv. Then R isQ6.
A binary relation R on \mathbb{N}\times \mathbb{N} is defined as follows: (a,b)R(c,d) if a \leq c or b \leq d. Consider the following propositions: P: R is reflexive Q: R is transitive Which one of the following statements is TRUE?Q7.
Let R_{1} be a relation from A =\left \{ 1,3,5,7 \right \} to B = \left \{ 2,4,6,8 \right \}. R_{2} be another relation from B to C = {1, 2, 3, 4} as defined below: i. An element x in A is related to an element y in B (under R_{1}) if x + y is divisible by 3. ii. An element x in B is related to an element y in C (under R_{2}) if x + y is even but not divisible by 3. Which is the composite relation R_{1}R_{2} from A to C?Q8.
Let R and S be any two equivalence relations on a non-empty set A. Which one of the following statements is TRUE?Q10.
Consider the binary relation: S = {(x, y)|y = x + 1 and x, y \in {0, 1, 2,...}} The reflexive transitive closure of S is