Announcements
ICS 6B } Regrades for Quiz #3 and Homeworks #4 & Boolean Algebra & Logic 5 are due Today
Lecture Notes for Summer Quarter, 2008
Michele Rousseau
Set 8 – Ch. 8.6, 11.6 Lecture Set 8 - Chpts 8.6, 11.1 2
Today’s Lecture
} Chapter 8 8.6 , Chapter 11 11.1 ● Partial Orderings 8.6 Chapter 8: Section 8.6 ● Boolean Functions 11.1
Partial Orderings (Continued)
Lecture Set 8 - Chpts 8.6, 11.1 3
What is a Partial Order? Some more defintions Let R be a relation on A. The R is a partial order iff R is: } If A,R is a poset and a,b are A, reflexive, antisymmetric, & transitive we say that: } A,R is called a partially ordered set or “poset” ● “a and b are comparable” if a b or b a } Notation: ◘ i.e. if a,b R and b,a R ● If A, R is a poset and a and b are 2 elements of A ● “a and b are incomparable” if neither a b nor b a such that a,b R, we write a b instead of aRb ◘ i.e if a,b R and b,a R } If two objects are always related in a poset it is called a total order, linear order or simple order. NOTE: it is not required that two things be related under a partial order. ● In this case A,R is called a chain. ● i.e if any two elements of A are comparable ● That’s the “partial” of it. ● So for all a,b A, it is true that a,b R or b,a R
5 Lecture Set 8 - Chpts 8.6, 11.1 6
1 Now onto more examples… More Examples Let A a,b,c,d and let R be the relation Let A 0,1,2,3 and on A represented by the diagraph Let R 0,0 1,1 , 2,0 , 2,2 , 2,3 3,3
The R is reflexive, but We draw the associated digraph: a b not antisymmetric a,c & c,a It is easy to check that R is 0 and not 1 c d transitive d,c c,a , but not d,a Refl. , antisym. & trans. 2 So A,R is a poset. 3 The elements 1,3 are incomparable because 1, 3 R and 3, 1 R so A,R is not a total ordered set.
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More Examples (3) Another example
Let A a,b,c,d and let R be the related Matrix: A all people R the relation “a not taller than b” 1 1 1 0 To check Then 0 1 1 0 properties ab 0 0 11 more easily R is reflexive a is not taller than a 1 1 0 1 lets convert it to a diagraph RiR is not antiiisymmetric [][ ] d c ● If “a is not taller than b” and “b is not taller than a”, then a and b have the same height, but a is not necessarily equal to b – it could be 2 people of the We see that R is reflexive loops at every vertex same height. and antisymmetric no double arrows Not a poset! It is not transitive eg. c,d & d,a , but no c,a
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Lexicographic Order
Lexicographic Order Suppose that A1. 1 and A2, 2 are two posets. } Lexicographic order is how we order words We construct a partial ordering on the Cartesian product A1x A2.
in the dictionary Given 2 elements a1, a2 and b1, b2 in A1x A2 ● It is AKA dictionary order or alphabetic order We say that a1, a2 b1, b2 Iff a b but a b ● First, we compare the 1st letters, if they are 1 1 1 1 or a b and a b . equal then we check the 2nd pair and etc. 1 1 2 2 ◘ E.g Let S 1,2,3 In other words,
◘ The lexicographical order of a1, a2 b1, b2 iff
P S , 1 , 1,2 , 1,2,3 , 1,3 , 2 , 2,3 , 3 1. the first entries of a1, a2 the first entries of b1, b2 or } We can apply this to any poset 2. the first entries of a1, a2 and b1, b2 , nd nd ● On a poset it is a natural order structure on the and the 2 entries of a1, a2 the 2 entries of b1, b2 entry Cartesian product. 11 12
2 Examples Example (2) Which of the following are true? EX. A1 A2 ; R1 R2 Then 1,3 1,4 ; 1,3 2,0 ; 3,5 4,8 True 1, 3 1, 3 4,4 2,8 False
In general, gg iven a1, a2 and b1, b2 , , 3,8 , 4,5 True compare a1 and b1 1,2,4,10 1,2,5,8 True If a b then 1 1 2,4,5 2,3,6 False a , a b , b 1 2 1 2 1,5,8 2,3,4 True ElseIf a1 b1 , then
If b1 b2 then
a1, a2 b1, b2 Lecture Set 8 - Chpts 8.6, 11.1 13 Lecture Set 8 - Chpts 8.6, 11.1 14
Strings Well Ordered Set } We apply this ordering to strings of symbols where } Let A,R be a poset there is an underlying 'alphabetical' or partial order which is a total order in this case . We say that A is a well‐ordered set if any non‐empty subset B A has a least element. Example: In other words, } Let A a, b, c and suppose R is the natural a B is a least element if a b for all b B. alphabetical order on A: recall that R is denoted by a R b and b R c. Example Then , is well ordered ● Any shorter string is related to any longer string comes ● Any subset B which is not empty has a least element. before it in the ordering . ● Eg if B 2,3,4,5,27,248,1253 then the least element is ● If two strings have the same length then use the induced a 2, because a b for all b B. partial order from the alphabetical order: aabc R abac , is not well ordered 15 ● Choose B , there is no least element 16
Hasse Diagram Example Construct the Hasse diagram of To every poset A,R we associate a Hasse diagram A 1, 2, 3 , R ● A graph that carries less info than the diagraph } Thus R 1,1 , 1,2 , 1,3 , 2,2 , 2,3 , 3,3 To construct a Hasse diagram: Step 2: 1 Construct a digraph representation of the poset Step 1:Draw the Digraph Remove loops A, R so that all arcs point up except the loops . With arrows ppgpointing up 2 Eliminate all loops 3 • R is reflexive – SO we know they are there 3 Eliminate all arcs that are redundant because of Step 3: Eliminate 2 transitivity redundant transitive arcs • Keep a,b and b,c Æ remove a,c 4 Eliminate the arrows at the ends of arcs since 1 everything points up. and it is antisymmetric so arrows Step 4: Remove arrows
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3 Example (2) Example (3) Construct the Hasse diagram of P a, b, c , } The elements of P a, b, c are } Given the Hasse diagram write down all the ordered pairs in R
a , b , c } Okay, so what do we know about this diagram a, b , a, c , b, c {a, b, c} c d } We know it is antisymmetric a, b, c • and that the arrows point up } Basically, it shows } {a, c} {a, b} {b, c} We know it is reflexive a hierarchy } We also know it is transitive {a} {b} b {c} } Now that we have the diagraph R is easy to find
a R a,a , a,b , a,c , a,d , b,b , b,c , b,d , c,c , d,d Lecture Set 8 - Chpts 8.6, 11.1 19 Lecture Set 8 - Chpts 8.6, 11.1 20
Maximal and Minimal Elements Example
Let S, be a poset } In this diagram An element a S is called maximal if there What is the maximal element? is no b such that a b. What is the minimal element?
In other words {a, b, c} MMiaxima l a is not less than any element in the poset. Similarly, Note: there can {a, c} {a, b} {b, c} An element B S is called minimal if there is be more than 1 minimal and {a} {b} no b such that b a. maximal element {c} in a poset These are easy to find in a Hasse diagram Minimal Because they are the “top” and “bottom” elements Lecture Set 8 - Chpts 8.6, 11.1 21 Lecture Set 8 - Chpts 8.6, 11.1 22
Example (2) Greatest and Least Elements a S is called the greatest element of poset 3 Maximal elements S, , if b a , b S } If there is only 1 maximal element then it Note: Every poset is the greatest. h1has 1 or more a S is called the least element of poset S, maximal elements and 1 or more , if a b , b S Minimal elements } If there is only 1 minimal element then it is the least 2 Minimal elements Note: The greatest and least may not exist
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4 Examples Upper and Lower Bounds Let S be a subset of A in the poset A, R . Greatest Let A S – be any subset No Greatest } An element b S is an upper bound for A if a b , a S Note:Not : B is not necessarilyn c ssarily in A
} b is a lower bound for A if b a , a S
Note: There can be more than 1 upper or lower bound No least least
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Example Example (2) A c,d Upper bound: } In this diagram g What is the upper bound for all subsets? Lower bound: What is the lower bound for all subsets? fg a {a, b, c} UUbdpper bound A a c d e Upper bounds: Note: Because {a, c} {a, b} {b, c} a,c,d,f,g we are dealing a b Lower bound: with subsets the {a} {b} bounds don’t {c} a necessarily Lower bound Note: It must be reachable have to be in the From all elements in the subset Lecturesubset Set 8 - Chpts 8.6, 11.1 27 Lecture Set 8 - Chpts 8.6, 11.1 28
Least Upper Bound & Example (2) Greatest Lower Bound A a,b,c h x S is called the least upper bound of A iff Upper bound: gf 1 – x is an upper bound for A e, f, h 2 – x is less than any other upper bound of A e Lower bound: d a x S is called the greatest lower bound of A iff b c 1 – x is an lower bound for A A a, c, d, f a 2 – x is greater than any other upper bound of A Upper bounds: h,f Lower bound: Note: These bounds may not exist if they do they are unique Lecture Set 8 - Chpts 8.6, 11.1 a 29 Lecture Set 8 - Chpts 8.6, 11.1 30
5 Example (2) Example (2) S, , | ‐‐‐ assume 0 A b, d, g A 3,9,12 ik Upper bound: Upper bounds: h g, h, I, k common multiples of 3,9,12 Ù Lower bound: gf b, a any positive integer which is divisible by 3, 9, 12 Ù e any positive integer which is divisible by 36 d Least upper bound: g Lower bounds: 3,1 Å common divisors of 3,9,12 b c Greatest lower bound: Least upper bound: b 36 least common multiple of 3,9,12 a Greatest lower bound: 3 greatest common divisor of 3,9,12 Lecture Set 8 - Chpts 8.6, 11.1 31 Lecture Set 8 - Chpts 8.6, 11.1 32
Lattices More Examples d Is this a lattice? A poset S, is called a lattice if every pair c e of elements a, b S has both a greatest what is the lub of b,c ?
lower bound and a least upper bound b None – not a lattice
a ElExample h Is this a lattice? Is this a lattice? Yes de g what is the glb of a,b ? f {a, b} e Is this a lattice? c None – not a lattice d {a} {b} Yes b c ab
a 33 Lecture Set 8 - Chpts 8.6, 11.1 34
Homework for 8.6
} 1 c‐d 3 a‐d ,5,7,9,11,19,21,23 b‐c , 25,33 a‐h ,35 a‐h ,43 a‐c Chapter 11: Section 11.1
Boolean Functions
Lecture Set 8 - Chpts 8.6, 11.1 35
6 Bit Operations Boolean Functions
n Let B={0,1}. Then B ={x1, x2,…, xn} | B for 1 i n is the Boolean Sum ‐ denoted by , OR, or set of all possible n‐tuples of 0s and 1s. The variable x is 1 0 1, 0 1 1, 1 1 1 called a Boolean variable if it assumes values only from B A function form Bn to B is called a Boolean Function of 0 0 0 Only time it =s 0 degree n. Boolean Product‐ ydenoted by · ,,, AND, or AlA Boolean varia ble x is a bit binary digit 1 · 1 1 Only time it =s 1 – accepts values 0 and 1. So x 0,1 . 1 · 0 0, 0 · 1 0, 0 · 0 0 _ A Boolean function of degree 1 is a function of Complement notated by , –NOT, or _ 1 Boolean variable with the values in 0,1 0 1_ F: 0,1 0,1 , x F x 1 0
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Examples x F x x Examples (2) For Example: 1 10 F1 x x A Boolean function of degree 2 is a function of two 01 variables with values in 0,1 x¬xF2 x x Example: xy F x,y x+y F x x 1 2 10 1 00 0 F1 x,y x y 01 0 01 1 10 1 x G(x)=x Notice these are = 11 1 G x x 11 xy F1 x,y xy 00 00 0 F1 x,y xy 01 0 If G(x) takes exactly the same value as F2. We say 10 0 11 1 F2 and G are the same function Even if they are defined by different expressions! Lecture Set 8 - Chpts 8.6, 11.1 39 Lecture Set 8 - Chpts 8.6, 11.1 40
Order of Operations
} In order to evaluate these we need to understand the order of preference ● Things in Parenthesis come first ● Then ◘ 1 – Complement ◘ 2 – Product ◘ 3 – Sum
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