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Notes

Relations

Some concepts

Questions

  1. State and prove De-Morgan’s law.
  2. Define partial ordering relation and equivalence relation.
  3. Represent the relation R = {(1,1), (2,1), (2,2), (2,3), (3,3)} defined on set A = {1, 2, 3} as the matrix form and directed graph.
  4. Construct the matrix and digraph representation of the relation R = {(a, b): a is multiple of b} defined on set A = {1, 2, 3, 4, 5, 6} .
  5. Prove that the relation R = {(a, b) : a, b ∈ L and a is parallel to b} is an equivalence relation, where L is the family of straight lines.
  6. Prove that the relation R = {(a, b) : a, b ∈ Z and (a - b) is divisible by 6} is an equivalence relation, where Z is the set of integers.
  7. Check the relation R = {(a, b) : a, b ∈ L and a is perpendicular to b} is an equivalence relation or not, where L is the family of straight lines.
  8. Prove that if R is an equivalence relation on set A then R-1 is also an equivalence relation on set A.
  9. Prove that the relation R = {(a, b) : a, b ∈ Z and a is less than or equal to b} is partial ordering relation, where Z is the set of integers.
  10. Let A = {2,3,5} and B = {6,8,10} and define a binary relation R from A to B as R = {(a, b) : a ∈ A, b ∈ B and a divides b} . Write each R and R-1 as a set of ordered pairs. Then find the domain and range for each R and R-1.
  11. Draw the Hasse diagram for the POSET [{1, 2, 3, 4, 6,12}, /] , where ‘/’ denotes “divide”. Then, find the maximal and minimal elements. Also find the minimum and maximum element if exist.
  12. Draw the Hasse diagram for the POSET A = {1, 2, 3, 4, 6, 8, 9,12,18, 24} under the relation “a divides b”. Then, find the maximal and minimal elements. Also find the minimum and maximum element if exist.
  13. Prove that the relation R = {(a, b) : (a - b) is divisible by 4 ∀ a, b ∈ Z } is an equivalence relation.
  14. Define the following: Composition of Relations, Indegree and outdegree of a graph.
  15. Draw the directed graph that represents the relation R = {(1,1), (2,2), (1,2), (2,3), (3,2), (3,1), (3,3)} on X = {1,2,3} .
  16. Represents the relation R = {(1,1), (1,3), (3,3), (4,4)} defined on set A = {1,3,4} in matrix form.
  17. If R is a relation defined on the set of natural numbers such that R = {(a, b) : a is divisible by b} then show that R is a partial order relation.
  18. Define the inverse of a relation. A relation R is defined from a set A to B such that a>b where a ϵ A and b ϵ B and A = {2,4,6} and B = {1,2,3} Find the relation R and R-1.
  19. Define poset and draw the Hasse diagram of ( P ( X ), ⊆) for X = {a, b, c} , where P(X) is the power set on X.