Each of the following defines a relation on N:
(i) x is greater than y,
(ii) x + y = 10,
(iii) x y is square of an integer
(iv) x + 4y = 10
Determine which of the above relations are reflexive, symmetric, and transitive.
(i)Here, x is greater than y;
If , then x > x, that does not satisfy for any .\\
Therefore, R is not reflexive.
Say,
For any , the above condition is not true.
Hence, R is not symmetric.
Again, xRy and yRz
Hence, R is transitive.
(ii) x + y = 10;
Thus,
Therefore,
So, R is not reflexive.
Again,
Therefore, R is symmetric.
And,
Therefore, R is not transitive.
(iii)Here, xy is square of an integer
So,
For any is an integer.
Thus, R is reflexive.
If
So, R is symmetric.
Again, if xy and yz both are square of an integer.
Then,
, this must be the square of an integer.
Therefore, R is transitive.
(iv) x + 4y = 10;
Hence, R is not symmetric.
Therefore, R is not transitive.