Let us break this problem down into four clear steps: finding the elements of relation R (k), finding the additions needed for reflexivity (m), finding the additions needed for symmetry (n), and computing the final sum.
Step 1: Find the elements of relation R and its count (k)
The set is given as A={−2,−1,0,1,2}.
The relation condition is:
y=max{x,1}
Let us compute the value of y for each element x∈A:
For x=−2: y=max{−2,1}=1⟹(−2,1)∈R
For x=−1: y=max{−1,1}=1⟹(−1,1)∈R
For x=0: y=max{0,1}=1⟹(0,1)∈R
For x=1: y=max{1,1}=1⟹(1,1)∈R
For x=2: y=max{2,1}=2⟹(2,2)∈R
Writing down the relation set explicitly:
R={(−2,1),(−1,1),(0,1),(1,1),(2,2)}
Count of elements in R:
k=5
Step 2: Find the minimum elements to add for Reflexivity (m)
A relation on set A is reflexive if every element relates to itself, meaning (x,x)∈R for all x∈A.
For our set A, a reflexive relation must contain the following pairs:
{(−2,−2),(−1,−1),(0,0),(1,1),(2,2)}
Looking at our current relation R, it already contains (1,1) and (2,2). The missing identity pairs that need to be added are:
{(−2,−2),(−1,−1),(0,0)}
Count of elements to be added for reflexivity:
m=3
Step 3: Find the minimum elements to add for Symmetry (n)
A relation is symmetric if whenever an ordered pair (x,y)∈R, its converse pair (y,x) must also belong to R.
Let us check each pair currently in R:
For (−2,1), we need (1,−2)
For (−1,1), we need (1,−1)
For (0,1), we need (1,0)
For (1,1), its reverse is (1,1), which is already present.
For (2,2), its reverse is (2,2), which is already present.
Therefore, the missing pairs that must be added to make R symmetric are:
{(1,−2),(1,−1),(1,0)}
Count of elements to be added for symmetry:
n=3
Step 4: Calculate k+m+n
Now substitute the values we computed into the final expression:
k+m+n=5+3+3
k+m+n=11
Conclusion
The value of k+m+n is 11.
Hence, the correct option is (a) 11.
