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Step-by-Step Solution
Step 1: Understand the Relation
We have the set
A = \{0,\,3,\,4,\,6,\,7,\,8,\,9,\,10\}
and a relation R on A such that
(x,y)\in R if either:
x - y is an odd positive integer, or
x - y = 2 .
We want to make R symmetric. A relation is symmetric if whenever (x,y)\in R , then (y,x)\in R as well.
Step 2: List All Ordered Pairs in R
Case I: x - y is an odd positive integer
For x - y to be an odd positive integer, x must be greater than y , and one of them must be odd while the other is even. Checking each possible pair in A under these conditions gives:
\{(3,0), (4,3), (6,3), (7,6), (7,4), (7,0), (8,7), (8,3), (9,8), (9,6), (9,4), (9,0), (10,9), (10,7), (10,3)\}
Number of such pairs = 15.
Case II: x - y = 2
We look for pairs (x,y) with x - y = 2 :
\{(6,4), (8,6), (9,7), (10,8)\}
Number of such pairs = 4.
Step 3: Determine Pairs Missing Their Symmetric Counterparts
For R to be symmetric, whenever (x,y) \in R , we must have (y,x) in R too. Currently:
All 15 pairs from the first case do not appear reversed in R (because reversing them would mean y-x is also an odd positive integer, which fails if x>y originally).
All 4 pairs from the second case do not appear reversed in R (reversing (x,y) giving (y,x) with y - x = -2 \neq 2 ).
Therefore, each of these 19 pairs is missing its reverse in the relation.
Step 4: Find the Minimum Number of Pairs to Add
We must add exactly the reverse of each pair in R that is missing. Since there are 19 pairs in R and none of them is symmetric on its own, the minimum number of new pairs needed is also 19 .
Step 5: State the Final Answer
The minimum number of elements that must be added to make R symmetric is
19.
