question_answer 1)

Determine whether each of the following relations are reflexive, symmetric and transitive : (i) Relation R in the set A = {1, 2, 3, … 13, 14} defined as R = {x, y} : 3x – y = 0} (ii) Relation R in the set N of natural numbers defined as R = {x, y) : y = x + 5 and x < 4} (iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x} (iv) Relation R in the set X of all integers defined as R = {(x, y) : x – y is an integer} (v) Relation R in the set A of human beings in a town at a particular time given by (a) R = {(x, y) : x and y work at the same place} (b) R = {(x, y) : x and y live in the same locality} (c) R = {(x, y) : x is exactly 7 cm taller than y} (d) R = {(x, y) : x is father of y} (e) R = {(x, y) : x is father of y}  

Answer:

(i)   A = {1, 2, 3, 4, ??. 13, 14:}               R = {(x, y) : 3x ? y = 0 i.e. y = 3x}                R = {(1, 3), (2, 6), (3, 9), (4, 12)}       Reflexive : As 1  A but (1, 1)  R.        R is not reflexive.       Symmetric : As (1, 3)  but (3, 1)        R is not symmetric.       Transitive : As (1, 3)  and (3, 9) but (1, 9) ,  R is not transitive.       Hence relation R is neither reflexive, nor symmetric, nor transitive.       (ii)    N = {1, 2, 3, 4, 5 6 ??}               R = {(x, y) : y = x + 5 and x < 4}                = {(1, 6), (2, 7), (3, 8)}       Reflexive : As  but (1, 1),  is not reflexive.       Symmetric : As (1, 6)  but (6, 1).       is not symmetric.       Transitive : Clearly R is transitive since it is not contradicted here.       Hence relation R is transitive but neither reflexive nor symmetric. (iii)   A = {1, 2, 3, 4, 5, 6}       R = {(x, y) : y is divisible by x}        R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6),       (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), (2, 6), (3, 6)} Reflexive : As (a, a)  is reflexive                               Symmetric : A (1, 2)  but (2, 1)  is not symmetric. Transitive : As (a, b)  and (b, c)  is divisible by a and c is divisible by b  is divisible by  (a, c)  is transitive. Hence relation R is reflexive, transitive but not symmetric. (iv) Z = {?.., ?3, ?2, ?1, 0, 1, 2, 3, ?}       R = {(x, y) : x ? y is an integer. Reflexive : As a ? a = 0 is an integer  is symmetric. Symmetric : As a ? b and b ? a are integers is symmetric.       Transitive : As a ? b and b ? c are integers and       (a ? b) + (b ? c) = a ? c is also an integer.       and (b, c)       is transitive.       Hence R is reflexive, symmetric and transitive.       (v) (a) Clearly R is reflexive, symmetry and transitive.                        (b) Clearly R is reflexive, symmetric and transitive.             (c) A = {x : x is human being in a town}             R = {(x, y) : x is exactly 7 cm taller than y}       Reflexive : As a is not 7 cm taller than a.              is not reflexive.       Symmetric : If a is exactly 7 cm taller than b, then b cannot be 7 cm taller, than a              R is not symmetric.       Transitive : If a exactly 7 cm taller than b and b is exactly 7 cm taller than c then a is exactly 14 cm taller than c.                      is not transitive.       Hence R is neither reflexive, nor symmetric nor transitive.       (d) A = {x : x is human being}             R = {(x, y) : x is a wife of y}       Reflexive : As a is not wife of              is not reflexive.       Symmetric : If a is a wife of b then b cannot be wife of a.             R is not symmetric.       Transitive : If a is a wife of b then b is a male ad a male cannot be a wife.        (a, b)        R is transitive as it is not contradicted here.        R is transitive but neither reflexive nor symmetric.       (e) Clearly R is neither reflexive nor symmetric nor transitive.