Q. 1E4.2( 114 Votes )

# Relation R in the

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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