Let 
. Let R be a relation on 
defined by 
if and only if 
. Let 
be the number of elements in R. Let 
and 
be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then 
is equal to
Text Solution
Verified by ExpertsB
To solve the problem, we start by defining the set 
and the relation 
on set 
, where an element 
is related to 
(written as 
) if and only if 
.
This leads us to the following pairs in the relation 
:
For 
, so 
is in 
.
For 
, so ( 
) is in 
.
For 
, so (0,1) is in 
.
For 
, so 
is in 
.
For 
, so 
is in 
.
Thus, the relation 
consists of the pairs:
and there are 
elements in R.
Making the Relation Reflexive
A relation is reflexive if every element in the set 
relates to itself. Therefore, the missing reflexive pairs are:
Adding these three pairs will make the relation reflexive, so 
.
Making the Relation Symmetric
A relation is symmetric if whenever ( 
) is in 
must also be in 
. Therefore, the missing symmetric pairs are:
Thus, we need to add these three pairs for symmetry, so 
.
Finally, we calculate the sum
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