Q6.Marks: +2.0UGC NET Paper 2: Computer Science 8th Oct 2022
Let ∈ = 0.0005, and Let Re be the relation {(x, y) = R2 ∶ |x − y| < ∈}, Re could be interpreted as the relation approximately equal. Re is
(A) Reflexive
(B) Symmetric
(C) transitive
Choose the correct answer from the options given below:
1.(A) and (B) only true✓ Correct
2.(B) and (C) only true
3.(A) and (C) only true
4.(A), (B) and (C) true
Solution
The correct answer: (A) and (B) only true.
A relation is said to be reflexive if every element in the set is related to itself. In this case, for any element x in R2, |x - x| = 0, which is less than ∈, meaning that (x, x) is in the relation Re. So Re is reflexive.
A relation is said to be symmetric if, for any ordered pair (x, y) in the relation, the ordered pair (y, x) is also in the relation. In this case, if (x, y) is in Re, meaning |x - y| < ∈, then |y - x| = |x - y| < ∈, meaning (y, x) is also in Re. So Re is symmetric.
However, Re is not transitive. Consider the following counterexample: let x = 0, y = 0.0004, and z = 0.0009. Then |x - y| = |0 - 0.0004| = 0.0004 < ∈, |y - z| = |0.0004 - 0.0009| = 0.0005 < ∈, but |x - z| = |0 - 0.0009| = 0.0009 ≠ |x - y| + |y - z|, meaning (x, z) is not in Re.
Therefore, Re is reflexive and symmetric but not transitive.