Notes
Relations
Questions
- State and prove De-Morgan’s law.
- Define partial ordering relation and equivalence relation.
-
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.
-
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} .
-
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.
-
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.
-
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.
-
Prove that if R is an equivalence relation on set A then R-1 is also an equivalence
relation on set A.
-
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.
-
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.
-
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.
-
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.
-
Prove that the relation R = {(a, b) : (a - b) is divisible by 4 ∀ a, b ∈ Z } is an
equivalence
relation.
-
Define the following:
Composition of Relations, Indegree and outdegree of a graph.
-
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} .
-
Represents the relation R = {(1,1), (1,3), (3,3), (4,4)} defined on set A = {1,3,4} in
matrix form.
-
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.
-
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.
-
Define poset and draw the Hasse diagram of ( P ( X ), ⊆) for X = {a, b, c} , where P(X)
is the power set on X.