Functions Question 473
Let n be a fixed positive integer. Define a relation R in the set Z of integers by aRb if and only if $ \frac{n}{a-b} \in \mathbb{Z}. $ The relation R is
Options:
A) reflexive
B) A equivalence relation
C) Transitive relation
D) Symmetrical
Show Answer
Answer:
Correct Answer: B
Solution:
A) Reflexive: For a relation to be reflexive, aRa must be true for all a in Z. aRa means n is divisible by (a-r), which is 0. But division by 0 is undefined, so the relation is not reflexive.
B) Symmetric: For a relation to be symmetric, if aRb then bRa must also be true. If aRb, then n is divisible by (a+b) If bRa, then n would need to be divisible by (b-a) But (b-a) = -(a-b), and if n is divisible by (a-b), it’s also divisible by -(a-b) So, the relation is symmetric.
C) Transitive: For a relation to be transitive, if aRb and bRc, then aRc must be true. aRb means a - b divides n bRc means n is divisible by (b+c) But this doesn’t necessarily mean that n is divisible by (a-c) So, the relation is not transitive.
D) Equivalence relation: For a relation to be an equivalence relation, it must be reflexive, symmetric, and transitive. We’ve shown it’s not reflexive and not transitive, so it’s not an equivalence relation.
Therefore, the correct answer is:
B) Symmetrical
The relation R is symmetric, but it’s neither reflexive, nor transitive, nor an equivalence relation.
BETA
