Consider the following relations on the set Z of all integers:
(A) \(R_1 = \{(x, y) : x \le y\}\)
(B) \(R_2 = \{(x, y) : x - y \text{ is divisible by } 4\}\)
(C) \(R_3 = \{(x, y) : |x - y| \le 1\}\)
(D) \(R_4 = \{(x, y) : x + y \text{ is even}\}\)
Which of the above are equivalence relations?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (B) and (D) only
A relation is an equivalence relation when it is reflexive, symmetric and transitive. One counter-example is enough to show that a property fails.
\(R_1\): it is reflexive and transitive, but not symmetric:
\(2 \le 5\) is true, but \(5 \le 2\) is false.
\(R_2\): all three properties hold:
\(x - x = 0\), which is divisible by 4
\(y - x = -(x - y)\), divisible by 4 whenever \(x - y\) is
\(x - z = (x - y) + (y - z)\), a sum of two multiples of 4
\(R_3\): it is reflexive and symmetric, but not transitive:
\(|0 - 1| = 1\) and \(|1 - 2| = 1\), but \(|0 - 2| = 2 \gt 1\)
\(R_4\): all three properties hold:
\(x + x = 2x\), which is even
\(y + x = x + y\)
\(x + z = (x + y) + (y + z) - 2y\), a sum and difference of even numbers
So \(R_2\) and \(R_4\) are equivalence relations, \(R_1\) is not symmetric and \(R_3\) is not transitive.
Checking the statements
(A) \(R_1 = \{(x, y) : x \le y\}\) — Incorrect. Not symmetric: \(2 \le 5\) but \(5 \not\le 2\).
(B) \(R_2 = \{(x, y) : x - y \text{ is divisible by } 4\}\) — Correct. Reflexive, symmetric and transitive.
(C) \(R_3 = \{(x, y) : |x - y| \le 1\}\) — Incorrect. Not transitive: (0, 1) and (1, 2) are in \(R_3\), but (0, 2) is not.
(D) \(R_4 = \{(x, y) : x + y \text{ is even}\}\) — Correct. Reflexive, symmetric and transitive.