Discrrete mathematics for computer science 12induction

Induction

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The Idea of Induction

Color the integers ≥ 0
0, 1, 2, 3, 4, 5, …
I tell you, 0 is red, & any int
next to a red integer is red,
then you know that

all the ints are red!
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Induction Rule

R(0)
and (" n)(R(n)fi R(n+1))
R(0),("
R(1m
), R(2),…,R(n),…
)R(m)

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Like Dominos…
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Example Induction Proof

Let’s prove:

2
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+
1
)
r
-1

r -1


Example Induction Proof
Base Case (n = 0):

? r 0+1 - 1

2
0
1+r +r +L +r =
r -1
1

OK!

r -1
=
=1
r -1


Example Induction Proof

• Inductive Step: Assume P(n) for some n ≥ 0
and prove P(n+1):

2

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 =
2

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 - 1 ÷+r


n+1
n+1
r - 1+r (r - 1)
This proves
=
r -1
P(n+1)
2

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:

2

1+r +r +L +r

n

• Can lead to confusion (n = 0?)
• Sum notation more precise

n

=∑ r
i=0

i