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