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How to Think About Algorithms by Jeff Edmonds

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Part Four Appendix

22 Existential and Universal Quantifiers

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Existential and universal quantifiers provide an extremely useful language for making formal statements. You must understand them. A game between a prover and a verifier is a level of abstraction within which it is easy to understand and prove such statements.

The Loves Example: Suppose the relation (predicate) Loves(p1, p2) means that person p1 loves person p2. Then we have

Expression Meaning Meaning
p2 Loves(Sam, p2) “Sam loves somebody.”
p2 Loves(Sam, p2) “Sam loves everybody.”
p1p2 Loves(p1, p2) “Somebody loves everybody.”
p1p2 Loves(p1, p2) “Everybody loves somebody.” ...

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