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.
E2. Show that the relation R in the set R of
real numbers, defined as R = {(a, b) : a 
} is neither
reflexive nor symmetric nor transitive.
Sol. R = {(a, b) : a 
a,
b, 
}
Reflexive : 
i.e. 
(a, a) 
Hence R is not reflexive.
Symmetric : As 2, 5 and
2 < 25 i.e. 2 < 52.
(2, 5)
But 
4 i.e. 
but (5, 2)
Hence R is not symmetric.
Transitive : As 3, ?2 and ?1 and
3 < (2)2 and ?2 < (?1)2.
and (?2,
?1)
But 
is not
symmetric.
Transitive : (a, b) and
(b, c) 
and 
is
transitive.
Hence R is reflexive and transitive but not
symmetric.
Example E 3. Check whether the relation R defined in the set {1,2, 3,4,
5, 6} as
R = {(a, b) : b = a + 1} is reflexive,
symmetric or transitive.
Sol. t A = {1, 2, 3,
4, 5. 6}
R = {(a,
b) : b = a + 1, a, b A}.
R =
{(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}.
Reflexive
: As 1 e A and (1, 1) R
R
is not reflexive.
Symmetric
: As 2, 3 A and (2, 3) R.
But
R
is not symmetric,
Transitive
: As 2. 3 and 4 A and (2, 3) R,
(3, t (2,4) R '
R
is not transitive.
Hence
R is neither reflexive, nor symmetric, nor transitive.
E 4. Show that the relation R defined
as R = {(a, b) : a 
b} is
reflexive and transitive
but not symmetric.
Sol. R
= {(a, b) : a b, a, b R
}
Reflexive
: As a a 
a
(a,
a) a
R
is reflexive.
Symmetric
: As 1, 2 and 1 2
(1,
2)
But
is
not symmetric.
Transitive
: (a, b) and (b, c) 
a,
b,c
and
R
is transitive.
Hence
R is reflexive and transitive but not
symmetric.
E5. Check whether the relation R in R defined
by R = {(a, b) : a 
} is
reflexive, symmetric or transitive.
Sol. R = {(a, b) : 
}
Reflexive : as 
i.e. 
is not
reflexive.
Symmetric : As 1 < 27 for 1, 3
i.e. (1, 3) for
1, 3
But 
is not
symmetric.
Transitive : As 100 < 125 and 5 <
8
i.e. 100 < 53 and 5 < 23
(100, 5) and
(5, 2)
But 100 
8 i.e. 100
< 23
(100, 2) 
(100, 5) ,
(5, 2) 
R is not
transitive.
Hence R is neither reflexive nor symmetric nor
transatie.
E6. Show that the
relation R is the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but
neither reflexive nor transitive.
Sol. Let A = {1, 2, 3} : R = {(1, 2), (2, 1)}
Reflexive : As 
and
(1, 1)
is not
reflexive.
Symmetric : As 1, 2 and
(1, 2)
Answer:
Let
y = f(x) 
for
every 
is correct.
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