Question

Is an empty set a function or not? Why or why not?

Answer 1

a function needs to have a variable x and y..an empty set doesnt have any elements so it's not a function

Answer 2

careful here

Answer 3

im wrong ??? o.O

Answer 4

a function does not need to have a variable \(x\) or some other variable \(y\)
definition of a function from A to B is usually defined as a particular kind of relation from A to B and a relation from A to B is a subset of \(A\times B\) with the restriction that no first coordinate appears twice

Answer 5

but isnt an empty set a "no solution"??

Answer 6

it is certainly counter intuitive, but many statements about the empty set are vacuously true. for example \(\emptyset \) is a subset of \(\emptyset\times \emptyset\)

Answer 7

so right away we know we have a relation. now is it also a function? answer is yes, because no first coordinate appears twice (since no first coordinate appears at all)

Answer 8

:O

well ill just silently accept my wrongliness ...:(

Answer 9

there is a lot of history behind the definition of functions, and it is still going on in some sense. originally the idea what that you had to have some sort of equation or a curve and it didn't have to pass the vertical line test, could just be any curve like \(x^2+y^2=4\)

Answer 10

later came the notion of one output for each input and still later came the idea that there did not need to be a formula of any kind, just a collection of pairs. in other words there did not need to be some method from getting from input to output, it could be random

Answer 11

that is why the modern definition takes a function to be a specific kind of relation, and a relation from a set A to a set B is defined as a subset of \(A\times B=\{(a,b):a \in A , b\in B\}\)