Solution
(i) Relation R in the set A = {1, 2,….,14} defined as R= {(x,y): 3x -y =0}
(a) Put y = x, 3x – x ≠ 0 => R is not reflexive.
(b) If 3x – y = 0, then 3y – x ≠ 0, R is not symmetric
(c) If 3x – y = 0, 3y – z = 0,then 3x – z ≠ 0,R is not transitive.
(ii) Relations in the set Nof natural numbers in defined by R = {(x, y): y = x + 5 and x < 4}
(a) Putting y = x, x ≠ x + 5, R is not reflexive
(b) Putting y = x + 5, then x ≠ y + 5,R is not symmetric.
(c) If y = x + 5, z = y + 5, then z ≠ x + 5 =>R is not transitive.
(iii) Relation R in the setA = {1,2,3,4,5,6} asR = {(x, y): y is divisible by x}
(a) Putting y = x, x is divisible by x => R is reflexive.
(b) If y is divisible by x, then x is not divisible by y when x ≠ y => R isnot symmetric.
(c) If y is divisible by x and z is divisible by y then z is divisible by xe.g., 2 is divisible by 1,4 is divisible by 2.
=> 4 is divisible by 1 => R is transitive.
(iv) Relation R in Z of allintegers defined as R = {(x, y): x – y is an integer}
(a) x – x=0 is an integer => R is reflexive
(b) x – y is an integer so is y – x => R is transitive.
(c) x – y is an integer, y- z is an integer and x – z is also an integer =>R is transitive.
(v) R is a set of humanbeings in a town at a particular time.
(a) R = {(x, y)} : x and y work at the same place. It is reflexive as x worksat the same place. It is symmetric since x and y or y and x work at same place.
It is transitive since X, y work at the same place and if y, z work at the sameplace,u then x and z also work at the same place.
(b) R : {(x, y) : x and y line in the same locality}
With similar reasoning as in part (a), R is reflexive, symmetrical andtransitive.
(c) R: {(x, y)}: x is exactly 7 cm taller than y it is not reflexive: x cannot7 cm taller than x. It is not symmetric: x is exactly 7 cm taller than y, ycannot be exactly 7 cm taller than x. It is not transitive: If x is exactly 7cm taller than y and if y is exactly taller than z, then x is not exactly 7 cmtaller than z.
(d) R = {(x, y): x is wife of y}
R is not reflexive: x cannot be wife of x. R is not symmetric: x is wife of ybut y is not wife of x.
R is not transitive: if x is a wife of y then y cannot be the wife of anybodyelse.
(e) R= {(x, y): x is a father of y}
It is not reflexive: x cannot be father of himself. It is not symmetric: x is afather of y but y cannot be the father of x.It is not transitive: x is a father of y and y is a father of z then x cannotbe the father of z.