Let \(A = \{1, 2, 3, \ldots, 9\}\) and let R be the relation on A given by \(R = \{(a, b) : a - b \text{ is even}\}\). Consider the following statements:
(A) R is an equivalence relation.
(B) \((2, 7) \in R\)
(C) R has exactly two equivalence classes.
(D) R contains 41 ordered pairs.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (C) and (D) only
\(a - b\) is even exactly when a and b are both odd or both even.
Reflexive: \(a - a = 0\) is even.
Symmetric: if \(a - b\) is even, so is \(b - a = -(a - b)\).
Transitive: \(a - c = (a - b) + (b - c)\) is a sum of two even numbers.
The classes are the odd and the even numbers of A:
\(\{1, 3, 5, 7, 9\}\), with 5 elements
\(\{2, 4, 6, 8\}\), with 4 elements
Every pair taken inside one class is in R, so the number of pairs is
\(5 \times 5 + 4 \times 4 = 25 + 16 = 41\)
So R is an equivalence relation with 2 classes, {1, 3, 5, 7, 9} and {2, 4, 6, 8}, and it has 41 ordered pairs.
Checking the statements
(A) R is an equivalence relation. — Correct. It is reflexive, symmetric and transitive.
(B) \((2, 7) \in R\) — Incorrect. \(2 - 7 = -5\) is odd.
(C) R has exactly two equivalence classes. — Correct. The odd numbers and the even numbers of A.
(D) R contains 41 ordered pairs. — Correct. \(25 + 16 = 41\).