Answer :

(a) It is given that R = {(x, y) : x and y work at the same place}

(x, x) ϵ R


R is reflexive.


Now, if (x, y) ϵ R, then x and y work on the same place.


y and x work at the same place.


(y, x) ϵ R


R is symmetric.


Now, let (x, y), (y, z) ϵ 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) ϵ R


R is transitive.


Therefore, R is reflexive, symmetric and transitive.


(b) It is given that R = {(x, y) : x and y live in the same locality}


(x,x) ϵ R as x and x live in the same human being.


R is reflexive.


Now, if (x,y) ϵ R, then x and y live in the same locality.


y and x live in the same locality.


(y,x) ϵ R


R is symmetric.


Now, let (x,y), (y,z) ϵ 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) ϵ R


R is transitive.


Therefore, R is reflexive, symmetric and transitive.


(c) It is given that R = {(x, y) : x is exactly 7 cm taller than y}


(x,x) R as human being x cannot be taller than himself.


R is not reflexive.


Now, if (x,y) ϵ R, then x is exactly 7 cm taller than y.


But y is not taller than x.


(y,x) R


R is not symmetric.


Now, let (x,y), (y,z) ϵ R


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.


Therefore, R is neither reflexive, nor symmetric, nor transitive.


(d) It is given that R = {(x, y) : x is wife of y}


(x,x) R as x cannot be the wife of herself.


R is not reflexive.


Now, if (x,y) ϵ R, then x is the wife of y.


But y is not wife of x.


(y,x) R


R is not symmetric.


Now, let (x,y), (y,z) ϵ R


x is the wife of y and y is the wife of z.


This cannot be possible.


(x,z) R


R is not transitive.


Therefore, R is neither reflexive, nor symmetric, nor transitive.


(e) It is given that R = {(x, y) : x is father of y}


(x,x) R as x cannot be the father of himself.


R is not reflexive.


Now, if (x,y) ϵ R, then x is the father of y.


But y is not father of x.


(y,x) R


R is not symmetric.


Now, let (x,y), (y,z) ϵ R


x is the father of y and y is the father of z.


x is not the father of z.


Indeed x is the grandfather of z.


(x,z) R


R is not transitive.


Therefore, R is neither reflexive, nor symmetric, nor transitive.


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