Lecture 2 part 2

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Lecture 2 (part 2)

• Partial Orders
• Reading Epp Chp 10.5

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Outline

• Revision
• Definition of poset
• Examples of posets
• In life
• Finite posets
• Infinite posets
• Notation
• Visualization Tool Hasse Diagram
• Definitions
• maximal, greatest, minimal, least.
• 2 Theorems
• 7. More Definitions
• Comparable, chain, total-order, well-order

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6. Some definitions
Given a partial order R (A,lt) and an element
xÎA,
(a) x is the maximal element iff "yÎA, x y
x y
(b) x is the greatest element iff "yÎA, y x
NOTE The text book defines maximal element in
another equivalent form. You may use that
definition. This definition in the lecture notes
should be easier when use in proofs.
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6. Some definitions
Given a partial order R (A, 9) and an element
xÎA,
(a) x is the maximal element iff "yÎA, x y
x y
(b) x is the greatest element iff "yÎA, y x
Expectation of Students When given a definition
using symbolic logic, you must not only be able
to understand the precise meaning (no more, no
less) but you must also try to extract out the
intuitive meaning behind the symbolic expression
(if such an extraction is possible). Take care
that you do not over simplify the meaning.
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6. Some definitions
Given a partial order R (A, ) and an element
xÎA,
(a) x is the maximal element iff "yÎA, x y
x y
No one is above me
(b) x is the greatest element iff "yÎA, y x
Im above everyone
Example 1
Q Are there any maximal elements?
A Yes, 3 and 1.
Q Are there any greatest elements?
A No.
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6. Some definitions
Given a partial order R (A,lt) and an element
xÎA,
(a) x is the maximal element iff "yÎA, x lt y
x y
No one is above me
(b) x is the greatest element iff "yÎA, yltx
Im above everyone
Example 2
Q Are there any maximal elements?
A Yes, 3.
Q Are there any greatest elements?
A Yes, 3.
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6. Some definitions
Given a partial order R (A,lt) and an element
xÎA,
(a) x is the maximal element iff "yÎA, xlty
x y
No one is above me
(b) x is the greatest element iff "yÎA, yltx
Im above everyone
Example 3
Q Are there any maximal elements?
A Yes, 4,8.
Q Are there any greatest elements?
A No.
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6. Some definitions
Given a partial order R (A,lt) and an element
xÎA,
(a) x is the maximal element iff "yÎA, xlty
x y
No one is above me
(b) x is the greatest element iff "yÎA, yltx
Im above everyone
(c) x is the minimal element iff "yÎA, yltx
x y
No one is below me
(d) x is the least element iff "yÎA, xlty
Im below everyone
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6. Some definitions
Q Are there any minimal elements?
A Yes, 2 and 4.
Q Are there any least elements?
A No.
(c) x is the minimal element iff "yÎA, yltx
x y
No one is below me
(d) x is the least element iff "yÎA, xlt y
Im below everyone
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6. Some definitions
Q Are there any minimal elements?
A Yes, 1.
Q Are there any least elements?
A Yes, 1.
(c) x is the minimal element iff "yÎA, yltx
x y
No one is below me
(d) x is the least element iff "yÎA, xlty
Im below everyone
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6. Some definitions
Given a partial order R (A,lt) and an element
xÎA,
(a) x is the maximal element iff "yÎA, xlty
x y
No one is above me
(b) x is the greatest element iff "yÎA, yltx
Im above everyone
(c) x is the minimal element iff "yÎA, yltx
x y
No one is below me
(d) x is the least element iff "yÎA, xlt y
Im below everyone
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6. Some definitions

• Thm A poset has at most one greatest element.
• Proof

I dont know what to prove. Well, Translate the
statement logically! Ok If m is the greatest
element and n is the greatest element, the m n.
Now do you know what to prove?
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6. Some definitions

• Thm A poset has at most one greatest element.
• Proof
• Let (A,lt) be a poset.
• Assume that m and n are greatest elements.
• Therefore m n

• Since m is the greatest element then (by
  definition) "yÎA, yltm.
• Which means that nltm.
• Since n is the greatest element then (by
  definition) "yÎA, yltn.
• Which means that mltn.
• Therefore nltm and mltn.
• (Since a poset is anti-symmetric).

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6. Some definitions

• Thm The greatest element (if it exists) is also
  the maximal element.
• Proof

Try again! Translate the statement
logically! Ok If m is the greatest element,
then m is the maximal element. Now do you know
what to prove?
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6. Some definitions

• Thm The greatest element (if it exists) is also
  the maximal element.
• Proof
• Let (A,lt) be a poset.
• Let m be the greatest element.
• Therefore m is the maximal element.

• Pick any element e.
• Assume that mlte.
• Since m is the greatest, then (by defn of
  greatest) "yÎA, yltm
• Which means that eltm.
• So we have mlte and eltm.
• Which means that m e.

(By defn of maximal "yÎA, mlty m y).
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Outline

• Revision
• Definition of poset
• Examples of posets
• In life
• Finite posets
• Infinite posets
• Notation
• Visualization Tool Hasse Diagram
• Definitions
• maximal, greatest, minimal, least.
• 2 Theorems
• 7. More Definitions
• Comparable, chain, total-order, well-order

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7. Some more definitions
Given a partial order R (A,lt) and let x,yÎA,
(a) x and y are comparable iff xlty or yltx
Example Given the following poset drawn as a
Hasse diagram,
Q Are 3 and 5 comparable?
A Yes.
Q Are 7 and 2 comparable?
A No.
Q Are 5 and 7 comparable?
A No.
Q Are 1 and 6 comparable?
A No.
Q Are 1 and 2 comparable?
A Yes.
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7. Some more definitions
Given a partial order R (A,lt) and let x,yÎA,
(a) x and y are comparable iff xlty or yltx
(b) Let B Í A. B is a chain iff "x,yÎB, x and
y are comparable.
Example Given the following poset drawn as a
Hasse diagram,
Q Is 3,5 a chain?
Q Is 1,2,4 a chain?
A Yes.
A Yes.
Q Is 1,7 a chain?
Q Is 3,5,6 a chain?
A Yes.
A No.
Q Is 1,4,6 a chain?
A No.
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7. Some more definitions
Given a partial order R (A,lt) and let x,yÎA,
(a) x and y are comparable iff xlty or yltx
(b) Let B Í A. B is a chain iff "x,yÎB, x and
y are comparable.
(c) If A is a chain, then R is a total order.
Example R (1,2,4,8,)
R (Z, )
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7. Some more definitions
Given a partial order R (A,lt) and let x,yÎA,
(a) x and y are comparable iff xlty or yltx
(b) Let B Í A. B is a chain iff "x,yÎB, x and
y are comparable.
(c) If A is a chain, then R is a total order.
(d) If R is a total order and R has a least
element, then R is well-ordered. (well-ordered
total order least element)
Example R (Z, )
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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

Hasse Diagram Draw it! Helps you see better
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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Proof of (1)
• Reflexive
• (m,n) R (m,n) is true,
• since (by defn of R) m m and (mm nn)

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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Proof of (1)
• Anti-symmetric
• Assume (m,n) R (p,q) and (p,q) R (m,n).
• Which means that (m,n) (p,q).

• Then (by defn of R) m p and (mp nq) and
  also p m and (pm qn)
• Therefore we know that m p and p m.
• Therefore mp.
• And since mp, then nq and qn.
• Therefore nq.
• And so mp and nq.
• (Defn of equality of pairs)

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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Proof of (1)
• Transitive
• Assume (m,n) R (p,q) and (p,q) R (r,s).
• Which means that (m,n) R (r,s).

• Then (by defn of R) m p and (mp nq) and
  also p r and (pr qs)
• Therefore we know that m p and p r.
  Therefore m r.
• Assume that mr
• Since we also know that m p and p r,
  therefore mpr.
• And since, (mp nq), and also mp, therefore
  nq.
• And since, (pr qs), and also pr, therefore
  qs.
• Which means that ns

• And so (m r) and (mr ns).

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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Proof of (2)
• Show that R is a total order.
• (DECODE THE DEFINITIONS!!!)
• which means that show that
• (1) R has a least element and
• (2) (Z)2 is a chain under R.
• Least element is (1,1).
• Prove " (p,q) Î (Z)2, (1,1) R (p,q)
• True since 1 p and (1p 1q)

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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Still need to show (Z)2 is a chain under R.
• which means that show that any two elements
  from (Z)2 is comparable.
• which means that we pick any two elements from
  (Z)2, say (a,b) and (c,d) and show (by defn of
  comparable) that either (a,b) R (c,d) or (c,d) R
  (a,b).
• NOW, YOU START PROVING

Proof of (2 - Continued)
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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Pick any two elements from (Z)2, say (a,b) and
  (c,d)
• Either a lt c or a c or c lt a.
• Case 1 (a lt c)
• Then (ac bd) is vacuously true
• And also a c will be true.
• And so (a,b) R (c,d).
• Case 3 (c lt a)
• Then (ca db) is vacuously true
• And also c a will be true.
• And so (c,d) R (a,b).

Proof of (2 - Continued)
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8. An example

• Let R be a relation on (Z)2, such that
• (m,n) R (p,q) iff m p and (mp nq)
• Prove that R is (1) a poset (2) well-ordered

• Case 2 (a c)
• Then a c and c a are both true.
• Now, either b lt d or b d or d lt b.
• Case (I) (b lt d)
• Then b d. So a c and (ac bd)
• Therefore (a,b) R (c,d).
• Case (II) (b d)
• So (a,b) (c,d).
• Therefore (a,b) R (c,d). (Poset is reflexive)
• Case (III) (d lt b)
• Then d b . So c a and (ca db)
• Therefore (c,d) R (a,b).

Proof of (2 - Continued Considering (a,b) and
(c,d))
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• Well-orderedness of a set is required for another
  proof technique called
• MATHEMATICAL INDUCTION
• End of Lecture