Mappings and Inverses: More Than One Left Inverse?

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SUMMARY

The discussion centers on the concept of mappings in mathematics, specifically addressing the existence of multiple left inverses for non-surjective functions. It is established that a mapping can indeed have more than one left inverse when it is not surjective, as different elements in the codomain can correspond to the same element in the domain. The participant clarifies that this does not contradict the definition of a mapping, which states that each element in the domain is associated with one and only one element in the codomain.

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  • Understanding of basic set theory
  • Familiarity with the definitions of injective and surjective functions
  • Knowledge of mappings and their properties
  • Concept of left and right inverses in functions
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  • Explore examples of mappings with multiple left inverses
  • Learn about the implications of non-surjective mappings in functional analysis
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Mathematicians, educators, and students studying advanced algebra or functional analysis who seek to deepen their understanding of mappings and their properties.

kidsmoker
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Hi,
just re-reading my lecture notes and there's a bit where it says

"The same mapping may have more than one left inverse (if it is not surjective) and more than one right inverse (if it is not injective)."

I can see how a mapping could have more than one right inverse. But how could it have more than one left inverse? This implies that you have one element from the domain being mapped to two different elements in the codomain. But my definition of a map is

"If A and B are sets then a mapping from A to B is a rule which associates to each element of A one and only one element of B".

Surely this is a contradiction?

Thanks for your help.
 
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The different left inverses are for different values of the elements in the codomain that don't have inverse because f is not onto.
 
Ah i see. Thanks!
 

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