0.4: Functions
- Page ID
- 85510
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Let \(X\) and \(Y\) be finite sets. A function \(f: X \rightarrow Y\) is:
| injective | (1-to-1) if it maps distinct inputs to distinct outputs. Formally: \(x \neq x^{\prime} \Rightarrow\) \(f(x) \neq f\left(x^{\prime}\right)\). If there is an injective function from \(X\) to \(Y\), then we must have \(|Y| \geqslant|X|\) |
| surjective | (onto) if every element in \(Y\) is a possible output of \(f\). Formally: for all \(y \in Y\) there exists an \(x \in X\) with \(f(x)=y\). If there is a surjective function from \(X\) to \(Y\), then we must have \(|Y| \leqslant|X|\). |
| bijective | (1-to-1 correspondence) if \(f\) is both injective and surjective. If there is a bijective function from \(X\) to \(Y\), then we must have \(|X|=|Y|\) |