1.6 The Pigeonhole Principle.¶
Whitman College. 1.6 The Pigeonhole Principle. Combinatorics and Graph Theory.
Cited by¶
1 citation across 1 artifact.
Primes¶
- Injectivity
- This refusal of collisions is the load-bearing property, sharply opposed to the many-to-one hashing and quotient patterns. The left-inverse implication. Because distinctness is preserved, the input is recoverable from its output on the image — re-identification and trace-back are possible in principle. The codomain-capacity constraint. The codomain must be at least as large as the domain along the distinguishing axes; an undersized codomain forces collisions by pigeonhole, and collisions (accidental exhaustion or adversarial attack) are the characteristic failure mode.
Supported in partVerified against the work's full text
Supplies the pigeonhole principle itself (more objects than boxes forces two into one box); it says nothing about injective functions, left inverses, hashing or adversarial collisions.
“Suppose that $n+1$ (or more) objects are put into $n$ boxes. Then some box contains at least two objects.”
- This refusal of collisions is the load-bearing property, sharply opposed to the many-to-one hashing and quotient patterns. The left-inverse implication. Because distinctness is preserved, the input is recoverable from its output on the image — re-identification and trace-back are possible in principle. The codomain-capacity constraint. The codomain must be at least as large as the domain along the distinguishing axes; an undersized codomain forces collisions by pigeonhole, and collisions (accidental exhaustion or adversarial attack) are the characteristic failure mode.
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