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Misalkan

\mathbb{R}

suatu himpunan bilagan riil dan

\Re

merupakan suatu rela...

Dec 19, 2023

Misalkan

\mathbb{R}

suatu himpunan bilagan riil dan

\Re

merupakan suatu relasi pada

\mathbb{R}

dimana untuk setiap

a, b \in \mathbb{R}, a \Re b

jika dan hanya jika |a-b|<2. Tunjukkan bahwa

\Re

bukan merupakan suatu relasi ekuivalen. (Petunjuk: periksa sifat transitif dari relasi tersebut)

Solution by Steps

step 1

To show that

\Re

is not an equivalence relation, we need to check if it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity

step 2

Check reflexivity: For any

a \in \mathbb{R}

, we need to have

a \Re a

. This means |a-a|<2, which simplifies to 0<2. This is true for all real numbers, so

\Re

is reflexive

step 3

Check symmetry: For any

a, b \in \mathbb{R}

, if

a \Re b

, then we must have

b \Re a

. Since |a-b|<2, it follows that |b-a|<2 because the absolute value function is symmetric. Thus,

\Re

is symmetric

step 4

Check transitivity: For any

a, b, c \in \mathbb{R}

, if

a \Re b

and

b \Re c

, then we must have

a \Re c

. This means |a-b|<2 and |b-c|<2. We need to check if |a-c|<2 always holds

step 5

Consider

a=0

b=1.5

, and

c=3

. We have |a-b|=|0-1.5|=1.5<2 and |b-c|=|1.5-3|=1.5<2, so

a \Re b

and

b \Re c

. However, |a-c|=|0-3|=3\not<2, so a \nRe c. This counterexample shows that

\Re

is not transitive

Answer

The relation

\Re

is not an equivalence relation because it is not transitive.

Key Concept

Equivalence Relation Properties

Explanation

A relation on a set is an equivalence relation if and only if it is reflexive, symmetric, and transitive. The relation

\Re

fails to be transitive, as shown by a counterexample.