Question

In text it says that a function is a collection of pairs of numbers with the following property: if (a,b) & (a,c) are both in the collection, then b=c; in other words, the collection must not contain two different pairs with the same first element. Now in an other text I found that if b is a range of the function then it could have several preimages or no preimages at all (preimage means the domain) now these two definitions are opposite to each other & I don't really know that which one is true :|

Answer 1

@experimentX any clue?

Answer 2

the first definition is from Spivak's Calculus & the second is from Kolmogorov's Analysis

Answer 3

A function is defined as a relation such that an element on pre-image can have only one element on range set.
now a is the preimage, then it can have only one element,

ie, f(a) = b and f(a) = c , if f is function then b = c.

Answer 4

on the other hand, some elements on the range set can have two or more than two preimages
for example: - f(x) = x^2 can have -2 and +2 for domain and +4 for range.

Answer 5

okkkkkkkkkkkkkkkkkk so both definitions are actually true

Answer 6

got it thanks buddy @experimentX

Answer 7

also let us consider two sets of integers