One-to-One Correspondence (Bijection)
Big Idea
Imagine you're at a school dance. If every student has exactly one dance partner, and no one is left without a partner, then there's a one-to-one correspondence between the group of students and the group of partners. This powerful concept helps us compare sets, understand functions, and even explore infinity!
Formal Definition
A one-to-one correspondence (also called a bijection) between two sets $A$ and $B$ is a function $f: A \to B$ such that:
① One-to-One (Injective)
Every element in $A$ maps to a different element in $B$.
$$\text{If } f(a_1) = f(a_2), \text{ then } a_1 = a_2$$
No two inputs share the same output.
② Onto (Surjective)
Every element in $B$ is mapped to by some element in $A$.
$$\text{For every } b \in B, \text{ there exists } a \in A \text{ such that } f(a) = b$$
No element in $B$ is left unpaired.