Midterm II Handout - such that P (1) is true, and whenever...

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MATH 74 MIDTERM 2 NOTE SHEET Theorem 1. A function f : X Y is invertible if and only if it is a bijection. Let N k = { x N : 1 x k } . Theorem 2. If X is a finite set, then #( X ) = k if and only if there is a bijection f : N k X . Theorem 3. If X and Y are finite sets, then so is X Y , and #( X Y ) #( X ) + #( Y ) . Moreover #( X Y ) = #( X ) + #( Y ) if and only if X Y = . Theorem 4. If X and Y are finite sets and f : X Y is an injection, then #( X ) #( Y ) . Theorem 5. If X and Y are finite sets and f : X Y is a surjection, then #( X ) #( Y ) . Theorem 6. If X and Y are finite sets and f : X Y is a bijection, then #( X ) = #( Y ) . Theorem 7 (Induction principle) . If P ( n ) , n = 1 , 2 , 3 ,... is a list of statements
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Unformatted text preview: such that P (1) is true, and whenever k is such that P ( k ) is true, then P ( k + 1) is true, Then P ( n ) is true for all n N . [It is often helpful to specify exactly what family of statements P ( n ) you are proving by induction.] You may assume I am familiar with the logic of proof by contradiction. If you write a proof by contradiction, make sure to say this somewhere in particular, specify where exactly the contradiction comes from....
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