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.
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