Equivalence Relation/Examples/Non-Equivalence/Sum is Integer

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Example of Relation which is not Equivalence

Let $\R$ denote the set of real numbers.

Let $\sim$ denote the relation defined on $\R$ as:

$\forall \tuple {x, y} \in \R \times \R: x \sim y \iff x + y \in \Z$

Then $\sim$ is not an equivalence relation.


Proof

We have that $\sim$ symmetric:

$x + y \in \Z \implies y + x \in \Z$

from Real Addition is Commutative.


But $\sim$ is non-reflexive:

$2 \cdotp 3 + 2 \cdotp 3 = 4 \cdotp 6 \notin \Z$

for example.


Also, $\sim$ is non-transitive:

\(\displaystyle 1 \cdotp 3 + 2 \cdotp 7\) \(=\) \(\displaystyle 4 \in \Z\)
\(\displaystyle 2 \cdotp 7 + 3 \cdotp 3\) \(=\) \(\displaystyle 6 \in \Z\)
\(\displaystyle 1 \cdotp 3 + 3 \cdotp 3\) \(=\) \(\displaystyle 4 \cdotp 6 \notin \Z\)

for example.


So $\sim$ is not an equivalence relation.

$\blacksquare$


Sources