- Relation R in the set A of human beings in a town at a particular time given by
- R = {(x, y) : x and y work at the same place}
- R = {(x, y) : x and y live in the same locality}
- R = {(x, y) : x is exactly 7 cm taller than y}
- R = {(x, y) : x is wife of y}
- R = {(x, y) : x is father of y}
(v)
(a) R = {(x, y) : x and y work at the equivalent place}
For reflexive: x and x can work at same spot (x, x) ∈ R
R is reflexive.
For symmetric: x and y work at same spot so y and x additionally work at same spot. (x, y) ∈ R and (y, x) ∈ R
R is symmetric.
For transitive: x and y work at same spot and y and z work at same spot, then, at that point x and z additionally work at same spot.
(x, y) ∈ R and (y, z) ∈ R then, at that point (x, z) ∈ R is transitive
Accordingly, R is reflexive, symmetric and transitive.
(b) R = {(x, y) : x and y live in the equivalent locality} (x, x) ∈ R => R is reflexive.
(x, y) ∈ R and (y, x) ∈ R => R is symmetric.
Once more,
(x, y) ∈ R and (y, z) ∈ R then, at that point (x, z) ∈ R => R is transitive. Consequently, R is reflexive, symmetric and transitive.
(c) R = {(x, y) : x is actually 7 cm taller than y} x can not be taller than x, so R isn’t reflexive.
x is taller than y then y can not be taller than x, so R isn’t symmetric.
Once more, x is 7 cm taller than y and y is 7 cm taller than z, then, at that point x can not be 7 cm taller than z, so R isn’t transitive.
Hence, R is neither reflexive, nor symmetric and nor transitive.
(d) R = {(x, y) : x is spouse of y}
x isn’t spouse of x, so R isn’t reflexive.
x is spouse of y yet y isn’t wife of x, so R isn’t symmetric.
Once more, x is spouse of y and y is wife of z then x can not be wife of z, so R isn’t transitive. In this manner, R is neither reflexive, nor symmetric and nor transitive.
(e) R = {(x, y) : x is father of y}
x isn’t father of x, so R isn’t reflexive.
x is father of y yet y isn’t father of x, so R isn’t symmetric.
Once more, x is father of y and y is father of z then x can’t be father of z, so R isn’t transitive. Consequently, R is neither reflexive, nor symmetric and nor transitive.