A relation R is defined on the set of integers as x R y iff (x + y) is even.…
2000
A relation R is defined on the set of integers as x R y iff (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
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Correct answer: C
Given relation:
xRy ⟺ (x+y) is even
Two integers have an even sum when both are:
even, or
odd
Thus, integers are divided into two groups:
Even integers
Odd integers
Checking properties:
Reflexive:
x+x=2x
is always even.
Symmetric:
If x+y is even, then y+x is also even.Transitive:
If x+y and y+z are even, then x and z have same parity, so x+z is even.
Hence, RRR is an equivalence relation.
Equivalence classes:
Set of all even integers
Set of all odd integers
Therefore, there are 2 equivalence classes.
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