📂Set Theory

Preimage of a Function

Definition ¹

For a function $f: X \to Y$ and $B \subset Y$, the set $f^{-1}(B): = \left\{ x \in X \ | \ f(x) \in B \right\}$ is called the preimage or inverse image of $B$ under $f$.

Explanation

Although the notation is similar, the definition alone does not let us say what relationship the inverse image has with the inverse function, and the two must not be confused.

Some people might find “inverse image” natural in Korean while feeling that “preimage” is more natural in English. This is because the Chinese character 逆, meaning “reverse,” fits well the notion of an inverse image as simply “where it came from,” whereas the word “inverse” evokes the inverse function as mentioned above, so people are consciously reluctant to use it. Of course, there are also simple reasons, such as “preimage” being easier to pronounce and therefore used often, or the prefix “pre-” being unfamiliar and therefore avoided.

Basic Properties

• [1] Empty set: $$ f ( \emptyset ) = \emptyset $$
• [2] Singleton set: $$ x \in X \implies f \left( \left\{ x \right\} \right) = \left\{ f(x) \right\} $$
• [3] Monotonicity: $$ A \subset B \subset X \implies f (A) \subset f(B) \\ C \subset D \subset Y \implies f^{-1} (C) \subset f^{-1} (D) \\ f(X) \subset Y \iff X \subset f^{-1} (Y) $$
• [4] Union: $$ f \left( \bigcup_{\gamma \in \Gamma} A_{\gamma} \right)= \bigcup_{\gamma \in \Gamma } f \left( A_{\gamma} \right) \\ f^{-1} \left( \bigcap_{\gamma \in \Gamma} A_{\gamma} \right) = \bigcap_{\gamma \in \Gamma } f^{-1} \left( A_{\gamma} \right) $$
• [5] Intersection: $$ f^{-1} \left( \bigcup_{\gamma \in \Gamma} A_{\gamma} \right)= \bigcup_{\gamma \in \Gamma } f^{-1} \left( A_{\gamma} \right) \\ f \left( \bigcap_{\gamma \in \Gamma} A_{\gamma} \right) {\color{red}\subset} \bigcap_{\gamma \in \Gamma } f \left( A_{\gamma} \right) $$
• [6] Difference: $$ f (A) \setminus f (B) \subset f (A \setminus B) \\ f^{-1} (C) \setminus f^{-1}(D) = f^{-1} (C \setminus D) $$

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Note especially in [5] and [6] that a function cannot preserve intersections as they are. For equality to hold, $f$ must be injective.

Unlike the concepts of bijection and inverse function, which one can only grow accustomed to through repetition even if understood only roughly, the inverse image needs to be learned as quickly and accurately as possible. If you gloss over the inverse image, your intuition for the null space in linear algebra immediately weakens, and if that leaves no room for further growth, the effect reaches all the way to abstract algebra. Since it has many properties different from the image of a function, do not simply assume it is just the opposite and move on; study it properly so that you know it for certain.