Exam 2 Review

8.2, 8.5, 8.6, 9.1-9.6

Thm. 1 for 2 roots, Thm. 2 for 1 rootTheorem 1: Let c1, c2 be elements of the real numbers.

Suppose r2-c1r –c2=0 has two distinct roots r1 and r2,

Then the sequence {a n} is a solution of the recurrence relation an = c1an-1 + c2 an-2

iff an=α1r1n+ α2r2

n for n=0, 1, 2… where α1 and α2 are constants.

----------------------------------------------------------------------------------Theorem 2: Let c1, c2 be elements of the real numbers.

Suppose r2-c1r –c2=0 has only one root r0 ,

Then the sequence {a n} is a solution of the recurrence relation an = c1an-1 + c2 an-2 iff an=α1r0

n+ α2 n r0n

for n=0, 1, 2… where α1 and α2 are constants.

Steps for Solving 2nd degree LHRR-K

For degree 2: the characteristic equation is r2-c1r –c2=0 (roots are used to find explicit formula)• Find characteristic equation• Find roots• Basic Solution: an=α1r1

n+ α2r2n where r1 and r2

are roots of the characteristic equation• Solve for α1,α2 to find solution• Prove this is the solution

8.2 examplesEx with two roots:Let an=7an-1 – 10 an-2 for n≥2; a0=2, a1=1• Find characteristic equation

• Find solution

-------------------------Ex with one root:an =8an-1 -16an-2 for n≥2; a0=2 and a1=20• Find characteristic equation

• Find solution

Ex: an=7an-1 – 10 an-2 for n≥2; a0=2, a1=1

• Prove the solution you just found is a solution

8.5 - unions

|A1 A2 A3|=∑|Ai| - ∑|Ai ∩ Aj| + |A1∩ A2 ∩ A3|

|A1 A2 A3 A4|=∑|Ai| - ∑|Ai ∩ Aj| + ∑ |Ai∩ Aj ∩ Ak| -

|A1∩ A2 ∩ A3∩ A4|

Sports

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8.6- optional

Let Ai=subset containing elements with property Pi

N(P1P2P3…Pn)=|A1∩A2∩…∩An|

N(P1’ P 2 ‘ P 3 ‘…Pn ‘)= number of elements with none of the

properties P1, P2, …Pn=N - |A1 A2 … An|=N- (∑|Ai| - ∑|Ai ∩ Aj| + …

+(-1)n+1|A1∩ A2 ∩…∩ An|)= N - ∑ N (Pi) + ∑(PiPj) -∑N(PiPjPk) +…

+(-1)n N(P1P2…Pn)

Sample applications

• Ex 1: How many solutions does x1+x2+x3= 11 have where xi is a nonnegative integer with x1≤ 3, x2≤ 4, x3≤ 6 (note: harder than previous > problems)

• Ex: 2: How many onto functions are there from a set A of 7 elements to a set B of 3 elements

• Ex. 3: Sieve- primes• Ex. 4: Hatcheck-- The number of derangements of a set

with n elements is • Dn= n![1 - ]• Derangement formula will be given.

9.1- Relations

Def. of Function: f:A→B assigns a unique element of B to each element of A

Def of Relation?

RSAT

A relation R on a set A is called:• reflexive if (a,a) R for every a A • symmetric if (b,a) R whenever (a,b) R for

a,b A• antisymmetric : (a,b) R and (b,a) R only if

a=b for a,b A• transitive if whenever (a,b) R and (b,c) R,

then (a,c) R for a,b,c A

RSAT

A relation R on a set A is called:• reflexive if aRa for every a A• symmetric if bRa whenever aRb for every a,bA• antisymmetric : aRb and bRa only if a=b for a,bA• transitive if whenever aRb and bRc, then aRc for

every a, b, cA

• Do Proofs of these****

Combining relations

R∩SRSR – SS – RS ο R = {(a,c)| a A, c C, and there exists b B

such that (a,b) R and (b,c) S}Rn+1=Rn ⃘ R

Thm 1 on 9.1• Theorem 1: Let R be a transitive relation on a

set A. Then Rn is a subset of R for n=1,2,3,…• Proof

• 8.2– not much on 8.2--just joins and projections

9.3• Representing relations R on A as both matrices

and as digraphs (directed graphs)• Zero-one matrix operations: join, meet,

Boolean product• MR R6 = MR5 v MR6

• MR5∩R6 = MR5 ^ MR6

• MR6 °R5 = MR5 MR6

9.4

• Def: Let R be a relation on a set A that may or may not have some property P. (Ex: Reflexive,…) If there is a relation S with property P containing R such that S is a subset of every relation with property P containing R, then S is called the closure of R with respect to P.

• Find reflexive and symmetric closures

Transitive closures

9.4: Theorem 1: Let R be a relation on a set A.There is a path of length n from a to b iff (a,b)Rn

--In examples, find paths of length n that correspond to elements in Rn

R*

• Find R*=

• Sample mid-level proofs:– R* is transitive

9.5 and 9.6

• Equivalence Relations: R, S, T• Partial orders: R, A, T

• (see definitions in other notes)

Definitions to thoroughly know & use

• a divides b• ab mod m• Relation• Reflexive, symmetric, antisymmetric, transitive

(not ones like asymmetric, from hw)• Equivalence Relation- RST• Partial Order- RAT• Comparable• Total Order

Definitions to apply

• You won’t have to state word for word, but may need to fill in gaps or apply definitions:– 8.2: outline for Thm. 1: Let c1, c2 be elements of the real

numbers. Suppose r2-c1r –c2=0 has two distinct roots r1 and r2,

Then the sequence {a n} is a solution of the recurrence relation an = ____________ iff

an= __________ for n=0, 1, 2… where______

(you fill in the gaps on Thm 1)

– Ch. 9: Maximal, minimal, greatest, least element

Thereoms to know well and use

• 9.1: Theorem 1: Let R be a transitive relation on a set A. Then Rn is a subset of R for n=1,2,3,…

• 9.4: Theorem 1: Let R be a relation on a set A.There is a path of length n from a to b iff (a,b)Rn • 9.4: Thm. 2: The transitive closure of a

relation R is R* =

Mid-level proofs to be able to do

• Prove that a given relation R, S, A, or T using the definitions– Ex: Show (Z+,|) is antisymmemetric– Ex: Show R={(a,b)|ab mod m} on Z+ is transitive

• Some basic proofs by induction• Let R be a transitive relation on a set A. Then Rn is a subset

of R for n=1,2,3,…• R* is transitive• Provide a counterexample to disprove that a relations is R,

S, A, or T– Ex: Show R={(1,1),(1,2),(1,4),(2,1),(2,2),(3,3),(4,),(4,4)} on

{1,2,3,4} is not transitive

Procedures to do• Represent relations as ordered pairs, matrices, or digraphs• Find Pxy and Jx and composite keys (sec 9.2)• Create relations with designated properties (ex: reflexive, but not

symmetric• Determine whether a relation has a designated property• Find closures (ex: reflexive, transitive)• Find paths and circuits of a certain length and apply section 9.4 Thm. 1• Calculate R∩S,RS,R – S,S – R,S ο R,Rn+1=Rn ⃘ R• Given R, describe an ordered pair in R3

• Given an equivalence R on a set S, find the partition… and vice versa• Identify examples and non-examples of eq. relations and of posets• Create and work with Hasse diagrams: max, min, lub,…