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Exam 2 Review

Exam 2 Review. 8.2, 8.5, 8.6, 9.1-9.6. Thm . 1 for 2 roots, Thm . 2 for 1 root. Theorem 1: Let c 1 , c 2 be elements of the real numbers. Suppose r 2 -c 1 r –c 2 =0 has two distinct roots r 1 and r 2 ,

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Exam 2 Review

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  1. Exam 2 Review 8.2, 8.5, 8.6, 9.1-9.6

  2. Thm. 1 for 2 roots, Thm. 2 for 1 root Theorem 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+ α2r2n for n=0, 1, 2… whereα1 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-2iff an=α1r0n+ α2 n r0n for n=0, 1, 2… whereα1 and α2 are constants.

  3. 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=α1r1n+ α2r2n where r1 and r2 are roots of the characteristic equation • Solve for α1,α2 to find solution • Prove this is the solution

  4. 8.2 examples Ex 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

  5. Ex: an=7an-1 – 10 an-2 for n≥2; a0=2, a1=1 • Prove the solution you just found is a solution

  6. 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|

  7. 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)nN(P1P2…Pn)

  8. 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. 9.1- Relations Def. of Function: f:A→B assigns a unique element of B to each element of A Def of Relation?

  10. 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

  11. 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,bA • antisymmetric : aRb and bRa only if a=b for a,bA • transitive if whenever aRb and bRc, then aRc for every a, b, cA • Do Proofs of these****

  12. Combining relations R∩S RS R – S S – R S ο R = {(a,c)| a A, c C, and there exists b B such that (a,b)  R and (b,c)  S} Rn+1=Rn⃘ R

  13. 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

  14. 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

  15. 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

  16. 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 inRn

  17. R* • Find R*= • Sample mid-level proofs: • R* is transitive

  18. 9.5 and 9.6 • Equivalence Relations: R, S, T • Partial orders: R, A, T • (see definitions in other notes)

  19. Definitions to thoroughly know & use • a divides b • ab mod m • Relation • Reflexive, symmetric, antisymmetric, transitive (not ones like asymmetric, from hw) • Equivalence Relation- RST • Partial Order- RAT • Comparable • Total Order

  20. 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

  21. 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* =

  22. 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)|ab 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

  23. 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,RS,R – S,S – R,S ο R,Rn+1=Rn⃘ 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,…

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