Math Problem Statement

Let R be an equivalence relation on a set A . Prove b € [a] = = [a] = [b]

Solution

Let RR be an equivalence relation on a set AA, which means that RR satisfies three properties:

  1. Reflexivity: aRaaRa for all aAa \in A.
  2. Symmetry: If aRbaRb, then bRabRa.
  3. Transitivity: If aRbaRb and bRcbRc, then aRcaRc.

Let [a][a] represent the equivalence class of aAa \in A, i.e.,

[a]={xAaRx}.[a] = \{ x \in A \mid aRx \}.

Proof that b[a]    [a]=[b]b \in [a] \iff [a] = [b]:

(1) Assume b[a]b \in [a]:

This means that aRbaRb, by the definition of the equivalence class. We want to show that [a]=[b][a] = [b].

  • Take any x[a]x \in [a]. By definition, aRxaRx. Since RR is symmetric, we know aRbaRb implies bRabRa.
  • Since RR is also transitive, from bRabRa and aRxaRx, it follows that bRxbRx. Hence, x[b]x \in [b].

Thus, we have shown that [a][b][a] \subseteq [b].

  • Now, take any y[b]y \in [b]. By definition, bRybRy. Since aRbaRb and RR is symmetric, bRabRa, and by transitivity, from aRbaRb and bRybRy, we get aRyaRy. Hence, y[a]y \in [a].

Thus, [b][a][b] \subseteq [a].

Since we have both [a][b][a] \subseteq [b] and [b][a][b] \subseteq [a], it follows that [a]=[b][a] = [b].

(2) Assume [a]=[b][a] = [b]:

If [a]=[b][a] = [b], then by definition of equivalence classes, b[a]b \in [a].

Conclusion:

We have shown that b[a]    [a]=[b]b \in [a] \iff [a] = [b]. Therefore, the statement b[a]    [a]=[b]b \in [a] \iff [a] = [b] is true.


Would you like more details or have any questions? Here are some related questions:

  1. What are some examples of equivalence relations in everyday life?
  2. How does an equivalence class partition a set?
  3. Can an element belong to more than one equivalence class?
  4. What are the implications of transitivity in equivalence relations?
  5. How can you apply equivalence relations in set theory and logic?

Tip: Equivalence classes group elements of a set that are indistinguishable under a given relation, essentially forming partitions of the set!

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Math Problem Analysis

Mathematical Concepts

Set Theory
Equivalence Relations
Equivalence Classes

Formulas

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Theorems

Properties of Equivalence Relations (Reflexivity, Symmetry, Transitivity)

Suitable Grade Level

Undergraduate Mathematics or Advanced High School