# Relation R in the

(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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