Check whether the relation R in R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive
Check whether the relation R in R defined as R = {(a, b): a ≤ b3} is reflexive, symmetric or transitive
Solution and Explanation
R = {(a, b): a ≤ b3}
It is observed that
∴ R is not reflexive.
Now,
(1, 2) ∈ R (as 1 < 23 = 8)
But,
(2, 1) ∉ R (as 23 > 1)
∴ R is not symmetric.
We have (3, \(\frac {3} {2}\)), (\(\frac {3} {2}\), \(\frac {6} {5}\)) ∈ R as 3<(\(\frac {3} {2}\))3 and \(\frac {3} {2}\)<(\(\frac {6} {5}\))3
But (3, \(\frac {6} {5}\)) ∉ R as 3>(\(\frac {6} {5}\))3
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
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Concepts Used:
Types of Relation
TYPES OF RELATION
Empty Relation
Relation is said to be empty relation if no element of set X is related or mapped to any element of X i.e, R = Φ.
Universal Relation
A relation R in a set, say A is a universal relation if each element of A is related to every element of A.
R = A × A.
Identity Relation
Every element of set A is related to itself only then the relation is identity relation.
Inverse Relation
Let R be a relation from set A to set B i.e., R ∈ A × B. The relation R-1 is said to be an Inverse relation if R-1 from set B to A is denoted by R-1
Reflexive Relation
If every element of set A maps to itself, the relation is Reflexive Relation. For every a ∈ A, (a, a) ∈ R.
Symmetric Relation
A relation R is said to be symmetric if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.
Transitive Relation
A relation is said to be transitive if, (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a, b, c ∈ A
Equivalence Relation
A relation is said to be equivalence if and only if it is Reflexive, Symmetric, and Transitive.