📂Set Theory

The Original Image of a Function

Definition ¹

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

Explanation

Though the notation is similar, one cannot say that the inverse image and the inverse function are related just by the definitions alone, and one should not confuse them.

Some people might find [preimage] more natural in English as opposed to how naturally “inverse image” sounds in Korean. This might be because the Chinese character for inverse, which simply means ‘where it came from,’ fits the concept of inverse image well. In contrast, the term [inverse] might remind one of inverse functions, hence people might consciously avoid using it. Of course, there are also simpler reasons like [preimage] being easier to pronounce, thus being used more often, and the prefix ‘pre-’ being unfamiliar, thus 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 set: $$ f (A) \setminus f (B) \subset f (A \setminus B) \\ f^{-1} (C) \setminus f^{-1}(D) = f^{-1} (C \setminus D) $$

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Note especially in [5], [6] that a function cannot preserve the intersection as it is. To satisfy equality, $f$ must be injective.

Unlike the concept of bijection and inverse function, which one can become familiar with through repetition, it’s necessary to learn about the inverse image quickly and accurately. Skimming over the concept of inverse image can lead to a lack of intuition about null spaces in linear algebra, which, if not addressed, can even affect abstract algebra. Since there are many properties different from the function’s range, it is not just the opposite; one must study properly to fully understand.