`S` is a relation over the set `R` of all real numbers and it is given by `(a ,\ b) in ShArra bgeq0` . Then, `S` is symmetric and transitive only reflexive and symmetric only (c) antisymmetric relation (d) an equivalence relation

Options

A. symmetric and transitive only
B. reflexive and symmetric only
C. a partial order relation
D. an equivalence relation

Answer

D

Explanation

Reflexivity: For any `a in R`, we have
`a^(2)=aa ge0 implies(a,a)in S`
Thus, `(a, a) in S` for all `a in R`.
So, S is a reflexive relation on R.
symmetry: Let `(a, b) in S`. Then,
`(a,b)inSimpliesabge0impliesbage0implies(b,a)inS`
Thus, `(a, b)inSimplies(b,a)inS" for all a, b"in R`.
So, S is a symmetric relation on R.
Transitivity: Let `a, b, c in R` such that
`(a, b) in S and (b, c) in R`
`implies ab ge 0 and bc ge 0`
implies a, b, c are of the same sign.
`implies ac ge 0`
`implies (a, c) in R`.
Thus, `(a,b) in S, (b, c) in S implies (a, c) in S`.
So, S is a transitive relation on R.
Hence, S is an equivalence relation on R.