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:
\(\ds 1 \cdotp 3 + 2 \cdotp 7\) | \(=\) | \(\ds 4 \in \Z\) | ||||||||||||
\(\ds 2 \cdotp 7 + 3 \cdotp 3\) | \(=\) | \(\ds 6 \in \Z\) | ||||||||||||
\(\ds 1 \cdotp 3 + 3 \cdotp 3\) | \(=\) | \(\ds 4 \cdotp 6 \notin \Z\) |
for example.
So $\sim$ is not an equivalence relation.
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $2$. Equivalence Relations: Exercise $1 \ \text {(ii)}$