Definition: Injective Function

Definition: A function f from the set A to the set B ($f:A\rightarrow B$) is injective if for all elements “a” and “b” in the set A, $f(a)=f(b)$ implies that a=b. //

This means that if you tell me that two elements in A get sent to the same element in B, and moreover if you tell me that this function is injective, then I immediately know that the two elements in A that you’re talking about are really the same element.  This is just a formal way of saying that “no two elements in A get sent to the same element in B”.  In our example of the school dance in lesson 7, the idea of injectivity was described as the scenario in which no two boys were dancing with the same girl.  We note that this in no way implies that every girl has a dance partner, but rather just that those girls who do have dance partners only have one.  This does, however, motivate the next definition.

For more on the notion of injectivity, check out lesson 7.

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