​ (i) A={1,2,3…13,14}R={(x,y):3x−y=0}∴R={(1,3),(2,6),(3,9),(4,12)}​ R is not reflexive since (1,1),(2,2)…(14,14)∈R.Also, R is not symmetric as (1,3)∈R, but (3,1)∈/R⋅[3(3)−1=0]Also, R is not transitive as (1,3),(3,9)∈R, but (1,9)∈/R.[3(1)−9=0]Hence, R is neither reflexive, nor symmetric, nor transitive.(ii) R={(x,y):y=x+5 and x<4}={(1,6),(2,7),(3,8)}It is seen that (1,1)∈/R.∴R is not reflexive.(1,6)∈RBut,(1,6)∈/R ∴R is not symmetric.Now, since there is no pair in R such that (x,y) and (y,z)∈R, then (x,z) cannot belong to R.∴R is not transitive.Hence, R is neither reflexive, nor symmetric, nor transitive.(iii) A={1,2,3,4,5,6}R={(x,y):y is divisible by x}We know that any number (x) is divisible by itself.⇒(x,x)∈R∴R is reflexive.Now,(2,4)∈R[ as 4 is divisible by 2 ]But,(4,2)∈/ R. [ as 2 is not divisible by 4] ∴R is not symmetric.∴z is divisible by x.⇒(x,z)∈R∴R is transitive.Hence, R is reflexive and transitive but not symmetric.(iv) R={(x,y):x−y is an integer }Now, for every x∈Z,(x,x)∈R as x−x=0 is an integer.∴R is reflexive.Now, for every x,y∈Z if (x,y)∈R, then x−y is an integer.⇒−(x−y) is also an integer.⇒(y−x) is an integer.∴(y,x)∈R∴R is symmetric.Now,Let (x,y) and (y,z)∈R, where x,y,z∈Z⇒(x−y) and (y−z) are integers.⇒x−z=(x−y)+(y−z) is an integer.∴(x,z)∈R :R is transitive.Hence, R is reflexive, symmetric, and transitive.(v) (a) R={(x,y):x and y work at the same place }⇒(x,x)∈R∴R is reflexive.If (x,y)∈R, then x and y work at the same place.⇒y and x work at the same place.⇒(y,x)∈R ∴R is symmetric.Now, let (x,y),(y,z)∈R⇒x and y work at the same place and y and z work at the same place.⇒x and z work at the same place.⇒(x,z)∈R∴R is transitive.Hence, R is reflexive, symmetric, and transitive.(b) R={(x,y):x and y live in the same locality }Clearly (x,x)∈R as x and x is the same human being.∴R is reflexive.If (x,y)∈R, then x and y live in the same locality.⇒y and x live in the same locality.⇒(y,x)∈R∴R is symmetric.Now, let (x,y)∈R and (y,z)∈R.⇒x and y live in the same locality and y and z live in the same locality.⇒X and z live in the same locality.⇒(x,z)∈R∴R is transitive.Hence, R is reflexive, symmetric, and transitive.(c) R={(x,y):x is exactly 7 cm taller than y}Now,(x,x)∈/RSince human being x cannot be taller than himself.∴R is not reflexive.Now, let (x,y)∈R.⇒x is exactly 7 cm taller than y.Then, y is not taller than x.∴(y,x)∈/RIndeed if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x. ∴R is not symmetric.Now,⇒x is exactly 7 cm taller than y and y is exactly 7 cm taller than z.⇒x is exactly 14 cm taller than z.∴(x,z)∈/R∴R is not transitive.Hence, R is neither reflexive, nor symmetric, nor transitive.(d) R={(x,y):x is the wife of y}Now,(x,x)∈/RSince x cannot be the wife of herself.∴R is not reflexive.Now, let (x,y)∈R⇒x is the wife of y.Clearly y is not the wife of x ∴(y,x)∈/R Indeed if x is the wife of y, then y is the husband of x.∴R is not transitive.Let (x,y),(y,z)∈R⇒x is the wife of y and y is the wife of z.This case is not possible. Also, this does not imply that x is the wife of z.∴(x,z)∈/R ∴R is not transitive.Hence, R is neither reflexive, nor symmetric, nor transitive.(e) R={(x,y):x is the father of y}(x,x)∈/R As x cannot be the father of himself.∴R is not reflexive.Now, let (x,y)∈R.⇒x is the father of y.⇒y cannot be the father of y.Indeed, y is the son or the daughter of y.∴(y,x)∈/R ∴R is not symmetric.Now, let (x,y)∈R and (y,z)∈R⇒x is the father of y and y is the father of z.⇒x is not the father of zIndeed x is the grandfather of z∴(x,z)∈/R∴R is not transitive.Hence, R is neither reflexive, nor symmetric, nor transitive.