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COURSENOTES
CS5034:
Models of Computation
Cli#0Bord A. Sha#0Ber
Department of Computer Science
Virginia Tech
Copyright c#0D1996
Forms of Proof
Proofs:
#0F Truth Tables
#0F Proof by Example
#7B Existence proof and disproving only
#0F Proof by Exhaustive Checking
#0F Direct Proof
#7B ad hoc
#7B chain of inference
#0F Indirect Proof
#7B Proof of Contrapositive
#7B Proof by Contradiction
#0F If and Only If
#0F Induction #28later#29
1
Set Notation
N: Natural Numbers
Z: Integers
Q: Rational Numbers
R: Real Numbers
If S =fa;b;cg, then
power#28S#29= f;;fag;fbg;fcg;
fa;bg;fa;cg;fb;cg;fa;b;cgg:
2
Sets, Bags and Tuples
Bag or Multiset: A collection of objects that
may contain a #0Cnite number of redundant
occurrences of elements.
#5Bh;h;g;u#5D
Tuples have n members #28tuple of length n#29
that appear in an order.
#3C12;R;9#3E
Three elements, the #0Crst is 12, the second is R,
the third is 9.
#0F Set: fh;u;g;hg=fh;u;gg=fu;g;hg:
#0F Bag:
#5Bh;u;g;h#5D = #5Bh;h;g;u#5D6=#5Bh;u;g#5D...

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