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## Sample Document Text

Logic, Sets
Section 1.1 Logic
Logic 2
Introduction
„ Applications of discrete mathematics:
„ Formal Languages (computer languages)
„ Digital Circuits
„ Compiler Design
„ Data Structures
„ Computability
„ Algorithm Design
„ Relational Database Theory
„ Complexity Theory (counting)
Logic 3
Example (counting)
„ The Traveling Salesman Problem, important
in
„ Circuit design
„ Many other CS problems
„ Given:
„ n cities c
1
, c
2
, . . . , c
n
„ Distance between city i and j, d
ij
„ Find the shortest tour.
Logic 4
Example (cont)
„ Assume a very fast PC:
„ 1 flop = 1 nanosecond = 10
-9
sec.
= 1,000,000,000 ops/sec = 1 GHz.
„ A tour requires n-1 additions. How many
different tours?
„ Choose the first city n ways,
the second city n-1 ways,
the third city n-2 ways,
etc.
Logic 5
Example (cont)
„ # tours = n (n-1) (n-2) . . . .(2) (1) = n! (Combinations)
Total number of additions = (n-1) n! (Rule of Product)
„ If n=8, T(n) = 7.8! = 282,240 flops < 1/...

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