Consider two sets. One is a singleton, consisting of the point "15". The other is the interval $[0,5]$. The mapping $15\to [0,5]$ is a vertical line.
According to the Wikipedia article for bijections, this is not a bijection because it must be that "each element of one set is paired with exactly one element of the other set".
Here, $15$ is paired with multiple elements.
Which of "one-to-one" and "onto" are violated though, and how?
I am conused because "one-to-one" says two elements in the domain cannot map to the same element in the codomain. But the domain here only has 1 element, so there cannot be a violation?
And "onto" says that every element in the co-domain is mapped to by at least one element in the domain. Here, every element in the codomain is mapped to..
Best Answer
The "one-to-one" and "onto" language presupposes that you're talking about a function. A function, by definition, assigns a single value to each element of the domain. Ordinarily, the word "mapping" is used synonymously with "function." It does not apply in this situation. You could call this a relation, but most definitely not a function.