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Question
R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .
Options
Reflexive and transitive
Reflexive and symmetric
Symmetric and transitive
an equivalence relation
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Solution
Reflexive and symmetric
Reflexivity: Let x∈R. Then,
x−x=0 < 1
⇒ |x−x | ≤ 1
⇒(x, x) ∈ R for all x ∈ Z
So, R is reflexive on Z.
Symmetry : Let (x, y)∈R. Then,
| x−y | ≤ 0
⇒ |−(y−x) | ≤ 1
⇒ | y−x | ≤ 1 [ Since |x−y|=|y−x| ]
⇒ (y, x) ∈ R for all x, y ∈ Z
So, R is symmetric on Z.
Transitivity : Let (x, y) ∈ R and ( y, z ) ∈ R. Then,
| x−y | ≤ 1 and | y−z | ≤ 1
⇒ It is not always true that |x−y| ≤ 1.
⇒ (x, z) ∉ R
So, R is not transitive on Z.
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