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A function $f: X \to Y$ is bijective if it is both injective and surjective:
$$|X| = |Y| \iff \exists \text{ bijection } f: X \to Y$$
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Current topic: bijections
A function $f: X \to Y$ is bijective if it is both:
- Injective: $f(a) = f(b) \implies a = b$
- Surjective: $\forall\, y \in Y,\; \exists\, x \in X$ such that $f(x) = y$
$$|X| = |Y| \iff \text{there exists a bijection } f: X \to Y$$
Exercise. Prove that the composition of two bijections is a bijection.
Hint: Let $f: A \to B$ and $g: B \to C$ be bijections. Show $g \circ f: A \to C$ is injective and surjective separately.
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