`R` is a relation on the set `Z` of integers and it is given by `(x ,\ y) in RhArr|x-y|lt=1.` Then, `R` is (a) reflexive and transitive (b) reflexive and symmetric (c) symmetric and transitive (d) an equivalence relation

Options

A. reflexive and transitive
B. reflexive and symmetric
C. symmetric and transitive
D. an equivalence relation

Answer

B

Explanation

For any `x in Z`, we have
`|x-x|=0le1`
`therefore |x-x|le1 " for all x"inZ`
`implies (x, x) in R " for all x"in Z`
implies R is reflexive on Z.
Let `(x, y) in R`. Then,
`|x-y|le1implies|y-x|le1implies(y,x)inR`
Thus, `(x,y) in Rimplies(y,x)inR`.
So, R is a symmetric relation on Z.
We observe that `(1, 0) in R` and `(0, -1) in R`, but `(1, -1) cancelin R`.
So, R is not a transitive relation on Z.