关于函数image的set builder notation解读及cardinality关系的技术问询

Hey there! Great question—this is a super common point of confusion when you’re first getting to grips with function images, so let’s unpack this clearly:

First, let’s break down the set-builder notation your textbook uses:
{y | y = f(x) for some x ∈ X}

The phrase "for some x ∈ X" here doesn’t mean we’re only picking a subset of X’s elements to map. Instead, it’s a way of saying: "y is in the image if there exists at least one x in X such that applying f to x gives y." In other words, we’re collecting every possible y value that f can produce when you feed it any element from X—all x in X are considered, but if multiple x’s map to the same y, that y only shows up once in the image (since sets don’t allow duplicate elements).

Now, to your question about cardinality (size) of the image compared to X:
The image’s cardinality can be less than or equal to X’s cardinality, but never greater. It all depends on the type of function f:

• If f is injective (one-to-one): Every x in X maps to a unique y—no two different x’s produce the same y. In this case, the image has exactly the same cardinality as X. For example, if f(x) = x + 3 mapping integers to integers, every x gives a unique y, so the image is all integers (same size as the original set).
• If f is not injective: Multiple x’s can map to the same y. Here, the image’s cardinality will be smaller than X’s. For example, take f(x) = x² with X = {-2, -1, 0, 1, 2}. The image is {0, 1, 4}, which has 3 elements—way fewer than X’s 5. Even with infinite sets, non-injective functions can lead to smaller cardinalities (or in some edge cases, equal size, but that’s a quirk of infinite sets).
• The extreme case: If f is a constant function (every x in X maps to the same y₀), the image is just {y₀}—cardinality 1, which is way smaller than X’s cardinality (as long as X has more than one element).

To clear up your confusion about set-builder notation: You’re right that the notation captures all possible y values that satisfy the condition. But the condition is about y’s existence (there’s some x that maps to it), not about preserving every x’s unique output. Sets automatically deduplicate elements, so even if 100 different x’s map to the same y, that y only appears once in the image.

So to wrap up: The image includes the output of f for every x in X, but duplicates are removed. Its size relative to X depends on whether f is one-to-one—if yes, same size; if no, smaller size.

备注：内容来源于stack exchange，提问作者Princess Mia