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Propositional Equivalences

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Propositional Equivalences CS/APMA 202, Spring 2005 Rosen, section 1.2 Aaron Bloomfield Tautology and Contradiction A tautology is a statement that is always true p ... – PowerPoint PPT presentation

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Title: Propositional Equivalences


1
Propositional Equivalences
  • CS/APMA 202, Spring 2005
  • Rosen, section 1.2
  • Aaron Bloomfield

2
Tautology and Contradiction
  • A tautology is a statement that is always true
  • p ? p will always be true (Negation Law)
  • A contradiction is a statement that is always
    false
  • p ? p will always be false (Negation Law)

p p ? p p ? p
T T F
F T F
3
Logical Equivalence
  • A logical equivalence means that the two sides
    always have the same truth values
  • Symbol is ?or ? (well use )

4
Logical Equivalences of And
  • p ? T p Identity law
  • p ? F F Domination law

p T p?T
T T T
F T F
p F p?F
T F F
F F F
5
Logical Equivalences of And
  • p ? p p Idempotent law
  • p ? q q ? p Commutative law

p p p?p
T T T
F F F
p q p?q q?p
T T T T
T F F F
F T F F
F F F F
6
Logical Equivalences of And
  • (p ? q) ? r p ? (q ? r) Associative law

p q r p?q (p?q)?r q?r p?(q?r)
T T T T T T T
T T F T F F F
T F T F F F F
T F F F F F F
F T T F F T F
F T F F F F F
F F T F F F F
F F F F F F F
7
Logical Equivalences of Or
  • p ? T T Identity law
  • p ? F p Domination law
  • p ? p p Idempotent law
  • p ? q q ? p Commutative law
  • (p ? q) ? r p ? (q ? r) Associative law

8
Corollary of the Associative Law
  • (p ? q) ? r p ? q ? r
  • (p ? q) ? r p ? q ? r
  • Similar to (34)5 345
  • Only works if ALL the operators are the same!

9
Logical Equivalences of Not
  • (p) p Double negation law
  • p ? p T Negation law
  • p ? p F Negation law

10
Sidewalk chalk guy
  • Source http//www.gprime.net/images/sidewalkchalk
    guy/

11
DeMorgans Law
  • Probably the most important logical equivalence
  • To negate p?q (or p?q), you flip the sign, and
    negate BOTH p and q
  • Thus, (p ? q) p ? q
  • Thus, (p ? q) p ? q

p q ?p ?q p?q ?(p?q) ?p??q p?q ?(p?q) ?p??q
T T F F T F F T F F
T F F T F T T T F F
F T T F F T T T F F
F F T T F T T F T T
12
Yet more equivalences
  • Distributive
  • p ? (q ? r) (p ? q) ? (p ? r)
  • p ? (q ? r) (p ? q) ? (p ? r)
  • Absorption
  • p ? (p ? q) p
  • p ? (p ? q) p

13
How to prove two propositions are equivalent?
  • Two methods
  • Using truth tables
  • Not good for long formula
  • In this course, only allowed if specifically
    stated!
  • Using the logical equivalences
  • The preferred method
  • Example Rosen question 23, page 27
  • Show that

14
Using Truth Tables
p q r p?r q ?r (p?r)?(q ?r) p?q (p?q) ?r
T T T T T T T T
T T F F F F T F
T F T T T T F T
T F F F T T F T
F T T T T T F T
F T F T F T F T
F F T T T T F T
F F F T T T F T
15
Using Logical Equivalences
Original statement
Definition of implication
DeMorgans Law
Associativity of Or
Re-arranging
Idempotent Law
16
Quick survey
  • I understood the logical equivalences on the last
    slide
  • Very well
  • Okay
  • Not really
  • Not at all

17
End of lecture on 25 January 2005
18
Logical Thinking
  • At a trial
  • Bill says Sue is guilty and Fred is innocent.
  • Sue says If Bill is guilty, then so is Fred.
  • Fred says I am innocent, but at least one of
    the others is guilty.
  • Let b Bill is innocent, f Fred is innocent,
    and s Sue is innocent
  • Statements are
  • s ? f
  • b ? f
  • f ? (b ? s)

19
Can all of their statements be true?
  • Show (s ? f) ? (b ? f) ? (f ? (b ? s))

b f s b f s s?f b?f b?s f?(b?s)
T T T F F F F T F F
T T F F F T T T T T
T F T F T F F T F F
T F F F T T F T T F
F T T T F F F F T T
F T F T F T T F T T
F F T T T F F T T F
F F F T T T F T T F
20
Are all of their statements true?Show values for
s, b, and f such that the equation is true
Original statement Definition of
implication Associativity of AND Re-arranging Idem
potent law Re-arranging Absorption
law Re-arranging Distributive law Negation
law Domination law Associativity of AND
21
What if it werent possible to assign such values
to s, b, and f?
Original statement Definition of implication ...
(same as previous slide) Domination
law Re-arranging Negation law Domination
law Domination law Contradiction!
22
Quick survey
  • I feel I can prove a logical equivalence myself
  • Absolutely
  • With a bit more practice
  • Not really
  • Not at all

23
Logic Puzzles
  • Rosen, page 20, questions 51-55
  • Knights always tell the truth, knaves always lie
  • A says At least one of us is a knave and B says
    nothing
  • A says The two of us are both knights and B
    says A is a knave
  • A says I am a knave or B is a knight and B says
    nothing
  • Both A and B say I am a knight
  • A says We are both knaves and B says nothing

24
Sand Castles
25
Functional completeness
  • Functional completeness is discussed on page 27
    (questions 37-39) of the text
  • All the extended operators have equivalences
    using only the 3 basic operators (and, or, not)
  • The extended operators nand, nor, xor,
    conditional, bi-conditional
  • Given a limited set of operators, can you write
    an equivalence of the 3 basic operators?
  • If so, then that group of operators is
    functionally complete

26
Rosen, 1.2 question 46
  • Show that (NAND) is functionally complete
  • Equivalence of NOT
  • p p ?p
  • ?(p ? p) ?p Equivalence of NAND
  • ?(p) ?p Idempotent law

27
Rosen, 1.2 question 46
  • Equivalence of AND
  • p ? q ?(p q) Definition of nand
  • p p How to do a not using nands
  • (p q) (p q) Negation of (p q)
  • Equivalence of OR
  • p ? q ?(?p ? ?q) DeMorgans equivalence of OR
  • As we can do AND and OR with NANDs, we can thus
    do ORs with NANDs
  • Thus, NAND is functionally complete

28
Quick survey
  • I felt I understood the material in this slide
    set
  • Very well
  • With some review, Ill be good
  • Not really
  • Not at all

29
Quick survey
  • The pace of the lecture for this slide set was
  • Fast
  • About right
  • A little slow
  • Too slow
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