How to interpret {y | y = f(x) for some x ∈ X}; does it mean this can have a lesser cardinality than X?

Question

My textbook defines the image of a set X for a function f to be {y | y = f(x) for some x ∈ X}. I was wondering what this definition says about how many elements of the set X are included in its image as f(x); would it be all of them, or perhaps a certain number of them less than its cardinality? Because it uses "some" rather than just saying x ∈ X, this had led me to consider the fact that this definition means the image of X might have a smaller cardinality than X. However, it seems in set builder notation our elements are reflective of all possibilities of the statement after | being eventualized, which makes me feel as if the image of X would have the same cardinality as X using this definition. So what would be the proper way to interpret this?

Answer 1

I understand the use of the word "some" is a source of confusion. Let us look at a function from the domain set $X$ to the codomain set $Y$.

Now the definition that sounds ambiguous to you will become precise once you interpret that as the qualifying condition for a $y$ of $Y$ to be included in the set (a subset of $Y$) called image of $X$.

It should be obtained as the value of the function $f$ for SOME input $x\in X$, it does not matter which $x$, hence the choice of the word some is significant.

A parallel. Which human beings are called MOTHERS? A person $y$ belonging to human race will be called a mother if there is SOME person x who was given birth to, by $y$.

Let H stand for the set of all human beings and M denote the function $M\colon H\to H$ defined as M(x) = mother of $x$.

The image of the function $M$ is precisely all the mothers.

Added in edit: Yes it is possible for the image to be of lesser cardinality than $X$. Because in the above example for siblings we have the same mother.