Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.
Solution and Explanation
R = {(a, b): a \(\leq\) b2}
It can be observed that \(\bigg(\frac{1}{2},\frac{1}{2}\bigg)\)∉ R, since \(\frac{1}{2}>\bigg(\frac{1}{2}\bigg)^2=\frac{1}{4}\)
∴R is not reflexive.
Now, (1, 4) ∈ R as 1 < 42
But, 4 is not less than 12.
∴(4, 1) ∉ R
∴R is not symmetric.
Now, (3, 2), (2, 1.5) ∈ R
(as 3 < 22 = 4 and 2 < (1.5)2= 2.25)
But, 3 > (1.5)2 = 2.25
∴(3, 1.5) ∉ R
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
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- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
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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.