Let \(Z_0\) be the set of all non-zero integers, and let R be the relation on \(Z_0\) defined by \(R = \{(a, b) : a = 2^{k} b \text{ for some integer } k\}\). Then R is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → an equivalence relation
Here k may be any integer: positive, zero or negative.
Reflexive: \(a = 2^{0} a\), so \((a, a) \in R\) with \(k = 0\).
Symmetric: if \(a = 2^{k} b\), then
\(b = 2^{-k} a\)
and \(-k\) is an integer, so \((b, a) \in R\).
Transitive: if \(a = 2^{k} b\) and \(b = 2^{m} c\), then
\(a = 2^{k} \cdot 2^{m} c = 2^{k + m} c\)
and \(k + m\) is an integer, so \((a, c) \in R\).
So R is an equivalence relation.