Determine whether each of the following relations are reflexive, symmetric and transitive: (i)Relation R in the setA= {1, 2, 3…13, 14} defined as R = {(x,y): 3x−y= 0} (ii) Relation R in the setNof natural numbers defined as R = {(x,y):y=x+ 5 andx< 4} (iii) Relation R in the setA= {1, 2, 3, 4, 5, 6} as R = {(x,y):yis divisible byx} (iv) Relation R in the setZof all integers defined as R = {(x,y):x−yis as integer} (v) Relation R in the setAof human beings in a town at a particular time given by (a) R = {(x,y):xandywork at the same place} (b) R = {(x,y):xandylive in the same locality} (c) R = {(x,y):xis exactly 7 cm taller thany} (d) R = {(x,y):xis wife ofy} (e) R = {(x,y):xis father ofy}

(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) ∈R But, (6, 1) ∉ 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. Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y. ∴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) ∉ R Since 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) ∉R Indeed if x is exactly 7 cm taller than y, then y is exactly 7 cm shorter than x. ∴R is not symmetric. Now, Let (x, y), (y, z) ∈ R. ⇒ x is exactly 7 cm taller thany 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) ∉ R Since 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 symmetric. 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 z. Indeed x is the grandfather of z. ∴ (x, z) ∉ R ∴R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive. read less