Math 2.0

Mathematical logic and foundations Definition

Equivalence relation induced by a function

Builds on FunctionEquivalence relation

Definition

Let f ⁣:A→Bf \colon A \to B be a function. The equivalence relation induced by ff is the relation ∼f\sim_f on AA with a∼fa′  ⟺  f(a)=f(a′)a \sim_f a' \iff f(a) = f(a') for all a,a′∈Aa, a' \in A.

Remarks

Another name is the kernel of ff. The relation ∼f\sim_f is an equivalence relation, because equality is one. Its classes are the nonempty preimages f−1({b})f^{-1}(\{b\}) of the points b∈Bb \in B. The function ff is injective exactly when ∼f\sim_f is equality.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionEquivalence relation induced by a function, definition, no evidence levelEquivalence relation …definition
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  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Existential quantifier definition
  8. Universal quantifier definition
  9. Cartesian product definition
  10. Subset definition
  11. Uniqueness quantifier definition
  12. Binary relation definition
  13. Reflexive relation definition
  14. Symmetric relation definition
  15. Transitive relation definition
  16. Function definition
  17. Equivalence relation definition
  18. Equivalence relation induced by a function definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Kernel_(set_theory), lead section
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MSC03E20
DefinesEquivalence relation induced by a function
UsesFunction, Equivalence relation

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