Relations Maths - Relations

The relation S defined on the set R of all real number by the rule aSb iff a ≥ b is:

1. An equivalence relation.
2. Reflexive, transitive but not symmetric.
3. Symmetric, transitive but not reflexive.
4. Neither transitive nor reflexive but symmetric.

The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is:

1. Symmetric only.
2. Reflexive only.
3. An equivalence relation.
4. Transitive only.

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m ∈ L. Then, R is:

1. Reflexive.
2. Symmetric.
3. Transitive.
4. None of these.

Give an example of a relation which is,
Symmetric and transitive but not reflexive.

Define a symmetric relation.

Write the domain of the relation R defined on the set Z of integers as follows:
(a, b) ∈ R ⇔ a² + b² = 25

If R = {(x, y): x + 2y = 8} is a relation on N by, then write the range of R.

State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive.

If A = {1, 2, 3, 4} define relations on A which have properties of being:
Reflexive, transitive but not symmetric.

Let A = {1, 2, 3}, and let R₁ = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}. Find whether or not the relations R₁ on A is:

1. Reflexive.
2. Symmetric.
3. Transitive.

Let A = {1, 2, 3}, and let R₃ = {(1, 3), (3, 3)}. Find whether or not the relations R₃ on A is:

1. Reflexive.
2. Symmetric.
3. Transitive.

Give an example of a relation which is,
Reflexive and symmetric but not transitive.

m is said to be related to n if m and n are integers and m - n is divisible by 13. Does this define an equivalence relation?

Show that the relation R, defined on the set A of all polygons as R = {(P₁, P₂): P₁ and P₂ have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

Show that the relation R on the set Z of integers, given by R = {(a, b): 2 divides a - b},  is an equivalence relation.

Let Z be the set of all integers and Z₀ be the set of all non-zero integers. Let a relation R on Z × Z₀ be defined as (a, b)R(c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z₀, Prove that R is an equivalence relation on Z × Z₀.