Injective
Definition 1. A function $f : X \longrightarrow Y$ is said to be injective or one-to-one if $x_1, x_2 \in X$ with $f(x_1) = f(x_2)$, then $x_1 = x_2$.
Surjective
Definition 2. A function $f : X \longrightarrow Y$ is said to be surjective or onto if $y \in Y$, then there exists $x \in X$ such that $f(x) = y$. In other words, $f : X \longrightarrow Y$ is surjective $\iff f(X) = Y$.
Bijective
Definition 3. A function $f : X \longrightarrow Y$ is said to be bijective if it is both injective and surjective. A bijection is also called a one-to-one correspondence.
Theorem 4에서 등호가 성립하지 않는 이유가 상수 함수 때문이라고 했는데, 그렇다면 상수 함수의 경우를 제외해 버린다면, 즉 injective의 조건을 걸어준다면 등호가 성립한다고 말할 수 있다.
Theorem 1
Theorem 1. Let $f : X \longrightarrow Y$ be a injection. Then $$f(\bigcap_{\gamma \in \Gamma} A_{\gamma}) \subseteq \bigcap_{\gamma \in \Gamma} f(A_{\gamma}).$$
Theorem 2
Theorem 2. Let $f : X \longrightarrow Y$ be a function. Then
(a) If there exists a function $g : Y \longrightarrow X$ such that $g \circ f = I_X$, then $f : X \longrightarrow Y$ is injective.
(b) If there exists a function $h : Y \longrightarrow X$ such that $f \circ h = I_Y$, then $f : X \longrightarrow Y$ is surjective.