Discrrete mathematics for computer science 12induction

Induction 2/24/12 The Idea of Induction Color the integers ≥ 0, 1, 2, 3, 4, 5, … I tell you, is red, & any int next to a red integer is red, then you know that all the ints are red! 2/24/12 Induction Rule R(0) and (" n)(R(n)fi R(n+1)) R(0),(" R(1m ), R(2),…,R(n),… )R(m) 2/24/12 Like Dominos… Click to edit Master text styles Second level Third level Fourth level Fifth level Example Induction Proof Let’s prove: 1+r +r + (for r ≠ 1) n +r = (n+ 1) r -1 r -1 Example Induction Proof Statements in magenta form a template for inductive proofs: • • Proof: (by induction on n) The induction hypothesis, P(n), is: 1+r +r2 +L +r n = (for r ≠ 1) ( n+ ) r -1 r -1 Example Induction Proof Base Case (n = 0): ? r 0+1 - 1+r +r +L +r = r -1 OK! r -1 = =1 r -1 Example Induction Proof • Inductive Step: Assume P(n) for some n ≥ and prove P(n+1): 1+r +r +L +r n+1 r = (n+1)+1 -1 r -1 Example Induction Proof Now from induction hypothesis P(n) we have r 1+r +r +L +r = so add r n n+1 n+1 -1 r -1 to both sides Example Induction Proof adding r n+1 to both sides,  r - 1 n+1 ( 1+r +r +L +r ) +r = r - ÷+r   n+1 n+1 r - 1+r (r - 1) This proves = r -1 P(n+1) n completing the proof by induction n+1 r = n+1 (n+1)+1 -1 r -1 an aside: ellipsis “” is an ellipsis Means you should see a pattern: 1+r +r +L +r n • Can lead to confusion (n = 0?) • Sum notation more precise n =∑ r i=0 i ... Example Induction Proof Let’s prove: 1+r +r + (for r ≠ 1) n +r = (n+ 1) r -1 r -1 Example Induction Proof Statements in magenta form a template for inductive proofs: • • Proof: (by induction... +r n = (for r ≠ 1) ( n+ ) r -1 r -1 Example Induction Proof Base Case (n = 0): ? r 0+1 - 1+r +r +L +r = r -1 OK! r -1 = =1 r -1 Example Induction Proof • Inductive Step: Assume P(n) for some