Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive. (ii) Transitive but neither reflexive nor symmetric. (iii) Reflexive and symmetric but not transitive. (iv) Reflexive and transitive but not symmetric. (v) Symm

Explanation

(i) Let set A = `{ 1, 2, 3}`
and relation R= `{(1, 2), (2, 1)]` is defined on A.
`because " " (1, 1) notin R`
`therefore R ` is not reflexive.
`because (1, 2) in R" " rArr (2, 1) in R`
` therefore R` is symmetic.
`because " " (1, 2) in R and (2,1 ) in R rArr (1, 1) notin R`
`therefore R` is not transitive.
(ii) Let a relation defined on the set of real numbers is `S = {(a, b): a lt b}`.
For each `a in R, (a, a) notin S` because ` a lt a` is false.
`therefore `S is not reflexive.
For each a, b, `in` R,
Let (a, b) `in S" " rArr a lt b`
`" " cancel(rArr ) b lt a `
`" " cancel (rArr) (b,a ) in S`
`therefore S` is not symmetric.
For each a, b, c `in` R
Let (a, b) `in S and (b, c) in S`
`rArr a lt b and blt c rArr a lt c rArr (a, c)in S `
`therefore S ` is transitive.
(iii) Let a realation defined on the set `A= {1, 2, 3}` is
`" " R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}`
`because ` For each `a in A, (a, a) in R`
`therefore R` is reflexive.
Each (a, b) `inR" " rArr(b,a ) in R`
`therefore R` is symmetric.
Each `(a, b) in R and (b,c ) in R cancel ( rArr ) (a, c) in R`
`therefore R` is not transitive.
`" " (because (1, 2) in R, (2, 3) in R ` but `(1, 3) notin R)`
(iv) Let a relation defined on the set of real numbers is
`" " R= {(a, b) : a^(3) ge b^(3)}`
For each ` a in R, (a, a) in R`
` therefore R` is reflexive.
If `(2,1) in R rArr 2^(3) gt 1^(3)`
`" " cancel (rArr ) 1^(3) gt 2^(3)`
`" " cancel (rArr) (1, 2) in R`
`therefore R` is not symmetric.
And each `(a, b) in R and (b, c) in R`
`rArr " " a^(3) ge b ^(3) and b^(3) ge c^(3)`
`rArr a^(3) ge c ^(3) `
`rArr (a, c) in R`
`therefore R` is transitive. (v) A relation defined on the set `A= {1, 2, 3}` is
`" " R= {(1, 1), (2,2), (1, 2), (2,1)}`
`because (3, 3) notin R`
`therefore R` is not reflexive.
Each `(a, b) in R rArr (b, a) in R`
`therefore R` is symmetric
Each ` (a,b) in R and (b, c) in R`
`rArr (a, c) in R`
`therefore R` is transititve.