Definition: A function is bijective if it is both injective and surjective. //
Thus, if you tell me that a function is bijective, I know that every element in B is “hit” by some element in A (due to surjectivity), and that it is “hit” by only one element in A (due to injectivity). In the example of the school dance from lesson 7, this means that every girl has a dance partner, and every girl has only one dance partner (perhaps the best scenario).
When there is a bijective function from the set A to the set B, we say that A and B are in a “bijective correspondence”, or that they are in a “one-to-one correspondence”. This latter terminology is used because each one element in A is sent to a unique element in B, and every element in B has a unique element in A assigned to it. We use the word “correspondence” in this case because a “bijection” is a two-way street. By this I mean that if I can find a bijective function from A to B, then I can also find one from B to A. All I do is simply reverse the “direction” of the function. I define a new function from B to A by taking each element “b” in B and asking which unique element in A got mapped to it by our initial bijective function, and then send “b” to that element. We know that this makes sense (i.e., satisfies the rules of a function) because every element in B had something sent to it (so our function sends all elements in B to something in A), and also because each element in B will only be assigned one element in A (so that the second requirement of a function—that everything in the domain is only sent to one element in the codomain—is satisfied).
For the original definition of a bijective function in the lessons, see lesson 7, and for a discussion of how our notion of bijectivity is important for counting and studying infinity, see lesson 8.