7/30/2019 Slide 2 1

1/3

( ,

Mathematics for Computer Science

MIT 6.042J/18.062J

Predicates & Quantifiers

Induction

opyright Albert R. Meyer, 2002. L2-1.1

Predicates

Predicates are

Propositions with variables

Example:

P(x,y) ::= x + 2 =y

is defined to be

Copyright Albert R. Meyer, 2002. L2-1.2

Predicates

2P x y) ::=x + = yx = 1 andy = 3: P(1,3) is true

x = 1 andy = 4: P(1,4) is false

P(1,4) is true

opyright Albert R. Meyer, 2002. L2-1.3

Quantifiers

x ForALLx

y There EXISTS somey

Copyright Albert R. Meyer, 2002. L2-1.4

+

Quantifiers

x,y range overDomain of Discourse

x y x

7/30/2019 Slide 2 1

2/3

r r

r r

Problems

Proof by InductionClass Problems 1& 2

opyright Albert R. Meyer, 2002. L2-1.7 Copyright Albert R. Meyer, 2002. L2-1.8

An Example of Induction

Suppose we have a property (say color) of

the natural numbers:

0, 1, 2, 3, 4, 5,

Showing thatzero is red, and that

thesuccessor of any red number is red,

proves that allnumbers are red!

opyright Albert R. Meyer, 2002. L2-1.9

The Induction Rule

0 and (from n to n+1)

proves 0, 1, 2, 3,.

R(0),n [R(n) R(n +1)]m ( ) R m

Copyright Albert R. Meyer, 2002. L2-1.10

Proof by Induction

Statements in green form a template for inductive

proofs:

Proof: (by induction on n)

The induction hypothesis:

rn+1

1P( ) ::= 1+ + 2 ++rn =

r1n

An Aside: Ellipses

Ellipses () mean that the reader is supposed

to infera pattern.

This can lead to confusion about what isbeing stated.

Here summation notation gives moreprecision, for example:

n

1+ + 2 ++rn = rii=0

Copyright Albert R. Meyer, 2002. L2-1.12opyright Albert R. Meyer, 2002. L2-1.11

7/30/2019 Slide 2 1

3/3

r r

r r

r r

Example Induction Proof An Example Proof

Base Case (n = 0):+ Revised Induction Hypothesis:

2 ++r0 ? r0 1

11+ + =

r1

P n) ::= 1 1+ + 2

++rn

=r

n

r

+1

1

1

1 ( rr1= =1

Wait: divide by zerobug!r1

This is only true forr1opyright Albert R. Meyer, 2002. L2-1.13 Copyright Albert R. Meyer, 2002. L2-1.14

An Example Proof An Example Proof

HaveP(n) by assumption:Induction Step: AssumeP(n) forn 0 to

1+ + 2 ++rn =rn+1 1proveP(n + 1): r1

n+r 1 1+r+r2 ++rn+1 =r( 1)+1 1 Adding rn+1 to both sides: n+1

r1 1+ + rn +rn+1 = r 1 +rn+1r1

1rn+1 + rn+1(r1)=r1

opyright Albert R. Meyer, 2002. L2-1.15 Copyright Albert R. Meyer, 2002. L2-1.16

L2-1.17opyright Albert R. Meyer, 2002.

An Example Proof

Continued

Which is justP(n+1)

QED.

1 11

( 1) 1

11

1

1

1

n nn

n

r r rr

r

r

r

+ ++

+ +

+ ++ + + =

=

1nn rr +

Problems

Class Problem 3

Copyright Albert R. Meyer, 2002. L2-1.18