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Section4.1The Pigeon Hole Principle

A function \(f:X\longrightarrow Y\) is said to be \(1\)–\(1\) (read one-to-one) when \(f(x)\neq f(x')\) for all \(x,x'\in X\) with \(x\neq x'\text{.}\) A \(1\)–\(1\) function is also called an injection or we say that \(f\) is injective. When \(f:X\longrightarrow Y\) is \(1\)–\(1\text{,}\) we note that \(|X|\le |Y|\text{.}\) Conversely, we have the following self-evident statement, which is popularly called the “Pigeon Hole” principle.

In more casual language, if you must put \(n+1\) pigeons into \(n\) holes, then you must put two pigeons into the same hole.

Here is a classic result, whose proof follows immediately from the Pigeon Hole Principle.

Proof

In Chapter 11, we will explore some powerful generalizations of the Pigeon Hole Principle. All these results have the flavor of the general assertion that total disarray is impossible.