Lead Section Page Testing

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   
Ans: The given relation is:
R = 1, 3 , 2, 6 , 3, 9 , 4, 12        
Since
1, 1 , 2, 2 …   
and
14, 14R .
We conclude that
R
is not reflexive.
Since
1, 3 R 
, but
3, 1 R 
. [since
3 3 -1 0   
]
We conclude that
R
does not belong to symmetric.
Since
1, 3
and
3, 9 R 
, but
1, 9 R. 3 1 -9 0        
.
We conclude that
R
is not transitive.
Therefore, the relation
R
is not reflexive, symmetric or transitive.
ii. Relation
R
in the set
N
of natural numbers defined as
R = x, y : y = x + 5 { 
and
x<4}
Ans: The given relation is:
R = 1, 6 , 2, 7 , 3, 8      .
Since
1, 1 R  .
We conclude that
R
is not reflexive.
Class XII Maths
Since
1, 6 R 
but
6, 1 R  .
We conclude that
R
does not belong to symmetric.
In the given relation
R
there is not any ordered pair such that
x, y
and
 y, z
both
R
, therefore we can say that
x, z
cannot belong to
R .
Therefore
R
is not transitive.
Hence, the given relation
R
is not reflexive, symmetric or transitive.