Let R be the relation on the set Z of all integers defined by \(R = \{(a, b) : 2a + 3b \text{ is divisible by } 5\}\). Then R is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → an equivalence relation
Reflexive: for every integer a,
\(2a + 3a = 5a\), which is divisible by 5.
Symmetric: let \(2a + 3b = 5m\) for some integer m. Then
\(2b + 3a = 5a + 5b - (2a + 3b)\)
\(2b + 3a = 5(a + b - m)\)
which is divisible by 5, so (b, a) is in R.
Transitive: let \(2a + 3b = 5m\) and \(2b + 3c = 5n\). Adding,
\(2a + 5b + 3c = 5m + 5n\)
\(2a + 3c = 5(m + n - b)\)
which is divisible by 5, so (a, c) is in R.
Hence R is an equivalence relation.