A relation R is defined on the set of integers as xRy if (x + y) is even.…
2000
A relation R is defined on the set of integers as xRy if (x + y) is even. Which of the following statements is true?
- A.
R is not an equivalence relation
- B.
R is an equivalence relation having 1 equivalence class
- C.
R is an equivalence relation having 2 equivalence classes
- D.
R is an equivalence relation having 3 equivalence classes
Attempted by 35 students.
Show answer & explanation
Correct answer: C
The relation is defined by xRy iff x + y is even. A sum x + y is even exactly when x and y have the same parity — both even or both odd.
Reflexive: for every integer x, x + x = 2x is even, so xRx holds.
Symmetric: if xRy then x + y is even; since y + x = x + y, yRx also holds.
Transitive: if xRy and yRz, then x, y share a parity and y, z share a parity, so x and z share a parity and x + z is even — hence xRz holds.
R is therefore an equivalence relation. It partitions the integers by parity into exactly two equivalence classes: the even integers and the odd integers. Hence the relation is an equivalence relation having 2 equivalence classes.