Equivalence Relation in Math: Proving Transitivity

Thread starter Dustinsfl
Start date Apr 9, 2010
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Equivalence Relation

In summary, an equivalence relation in math is a relationship between two or more elements in a set that satisfies the properties of reflexivity, symmetry, and transitivity. Proving transitivity in an equivalence relation means showing that if A is related to B and B is related to C, then A is also related to C. This is important in ensuring consistency and making logical deductions. To prove transitivity, one must provide a logical argument or proof using definitions, properties, and logic. Examples of proving transitivity include the "equal to" relation and the "parallel to" relation in geometry.

Apr 9, 2010

Dustinsfl

[tex]\forall a,b\in \mathbb{Z}[/tex]

[tex]a\sim b[/tex] iff. [tex]\left\vert a-b \right\vert \leq 3[/tex]

I have already shown reflexive and symmetric but not sure on how to show transitive.

I know the definition.

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Apr 9, 2010

Dick

Science Advisor

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Why don't you try to show it's not transitive?

FAQ: Equivalence Relation in Math: Proving Transitivity

What is an equivalence relation in math?

An equivalence relation is a mathematical concept that describes a relationship between two or more elements in a set. It is a type of relation that satisfies three properties: reflexivity, symmetry, and transitivity.

What does it mean to prove transitivity in an equivalence relation?

To prove transitivity in an equivalence relation, you need to show that if element A is related to element B, and element B is related to element C, then element A is also related to element C. In other words, the relation between element A and C is the same as the relation between A and B and B and C.

Why is proving transitivity important in an equivalence relation?

Proving transitivity is important because it ensures that the relation being defined is consistent and well-defined. It also allows us to make logical deductions and draw conclusions from the given relations.

How do you prove transitivity in an equivalence relation?

To prove transitivity, you need to provide a logical argument or proof that shows that the three elements in question are related in a consistent and transitive manner. This can involve using definitions, properties, and logic to show that the relation holds true.

What are some examples of proving transitivity in an equivalence relation?

One example is proving the transitivity of the "equal to" relation. If A = B and B = C, then A = C. Another example is proving the transitivity of the "parallel to" relation in geometry. If line A is parallel to line B and line B is parallel to line C, then line A is also parallel to line C.