lec-4.pdf

# lec-4.pdf - Today Today Comment Add 0 Today Comment Add 0...

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Today. Comment: Add 0. Add ( k - k ) . Induction: Some quibbles.
Today. Comment: Add 0. Add ( k - k ) . Induction: Some quibbles. Induction and Recursion

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Today. Comment: Add 0. Add ( k - k ) . Induction: Some quibbles. Induction and Recursion Couple of more induction proofs.
Today. Comment: Add 0. Add ( k - k ) . Induction: Some quibbles. Induction and Recursion Couple of more induction proofs. Stable Marriage.

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Some quibbles. The induction principle works on the natural numbers.
Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) .

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Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes.
Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3?

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Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n )
Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n ) Restate as:

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Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n ) Restate as: n N , Q ( n ) where Q ( n ) = ( n 3 ) = P ( n ) ”.
Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n ) Restate as: n N , Q ( n ) where Q ( n ) = ( n 3 ) = P ( n ) ”. Base Case: typically start at 3.

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Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n ) Restate as: n N , Q ( n ) where Q ( n ) = ( n 3 ) = P ( n ) ”. Base Case: typically start at 3. Since n N , Q ( n ) = Q ( n + 1 ) is trivially true before 3.
Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n ) Restate as: n N , Q ( n ) where Q ( n ) = ( n 3 ) = P ( n ) ”. Base Case: typically start at 3. Since n N , Q ( n ) = Q ( n + 1 ) is trivially true before 3. Can you do induction over other things? Yes.

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Some quibbles. The induction principle works on the natural numbers. Proves statements of form: n N , P ( n ) . Yes. What if the statement is only for n 3? n N , ( n 3 ) = P ( n ) Restate as: n N , Q ( n ) where Q ( n ) = ( n 3 ) = P ( n ) ”.
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