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Logic

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Section 1.1 Logic – PowerPoint PPT presentation

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Title: Logic


1
Section 1.1
  • Logic

2
Proposition
  • Statement that is either true or false
  • cant be both
  • in English, must contain a form of to be
  • Examples
  • Cate Sheller is President of the United States
  • CS1 is a prerequisite for this class
  • I am breathing

3
Many statements are not propositions ...
  • Give me liberty or give me death
  • ax2 bx c 0
  • See Spot run
  • Who am I and why am I here?

4
Representing propositions
  • Can use letter to represent proposition think of
    letter as logical variable
  • Typically use p to represent first proposition, q
    for second, r for third, etc.
  • Truth value of a proposition is T (true) or
    F(false)

5
Negation
  • Logical opposite of a proposition
  • If p is a proposition, not p is its negation
  • Not p is usually denoted
  • ?p

6
Truth table
  • Graphical display of relationships between truth
    values of propositions
  • Shows all possible values of propositions, or
    combinations of propositions

p ?p T F F T
7
Logical Operators
  • Negation is an example of a logical operation
    the negation operator is unary, meaning it
    operates on one logical variable (like unary
    arithmetic negation)
  • Connectives are operators that operate on two (or
    more) propositions

8
Conjunction
  • Conjunction of 2 propositions is true if and only
    if both propositions are true
  • Denoted with the symbol ?
  • If p and q are propositions, p ? q means p AND q
  • Remember - ? looks like A for And

9
Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
10
Truth table for p ? q
p q p ? q T T T T F F F T F F F
F
11
Disjunction
  • Disjunction of two propositions is false only if
    both propositions are false
  • Denoted with this symbol ?
  • If p and q are propositions, p ? q means p OR q
  • Mnemonic ? looks like OAR in the water (sort of)

12
Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
13
Truth table for disjunction
p q p ? q T T T T F T F T T F F
F
14
Inclusive vs. exclusive OR
  • Disjunction means or in the inclusive sense
    includes the possibility that both propositions
    are true, and can be true at the same time
  • For example, you may take this class if you have
    taken Calculus I or you have the instructors
    permission - in other words, you can take it if
    you have either, or both

15
Exclusive OR
  • The exclusive or of two propositions is true when
    exactly one of the propositions is true, false
    otherwise
  • Exclusive or is denoted with this symbol ?
  • For p and q, p ? q means p XOR q
  • Mnemonic ? looks like sideways X inside an O

16
English examples
  • I am either in class or in my office
  • The meal comes with soup or salad
  • You can have your cake or you can eat it

17
Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
18
Truth table for ?
p q p ? q T T F T F T F T T F F
F
19
Implication
  • The implication of two propositions depends on
    the ordering of the propositions
  • The first proposition is calls the premise (or
    hypothesis or antecedent) and the second is the
    conclusion (or consequence)
  • An implication is false when the premise is true
    but the conclusion is false, and true in all
    other cases

20
Implication
  • Implication is denoted with the symbol ?
  • For p and q, p ? q can be read as
  • if p then q
  • p implies q
  • q if p
  • p only if q
  • q whenever p
  • q is necessary for p
  • p is sufficient for q
  • if p, q

21
Implication
  • Note that a false premise always leads to a true
    implication, regardless of the truth value of the
    conclusion
  • Implication does not necessarily mean a cause and
    effect relationship between the premise and the
    conclusion

22
Implications in English
  • If Cate lives in Iowa, then Discrete Math is a
    3-credit class
  • Since p (I live in Iowa) and q (this is a
    3-credit class) are both true, p ? q is true even
    though p and q are unrelated statements

23
Implications in English
  • If the sky is brown, then 225
  • Since p (sky is brown) and q (225) are both
    false, the implication p ? q is true
  • Remember, you can conclude anything from a false
    premise

24
If/then vs. implication
  • In programming, the if/then logic structure is
    not the same as implication, though the two are
    related
  • In a program, if the premise (if expression) is
    true, the statements following the premise will
    executed, otherwise not
  • There is no conclusion, so its not an
    implication

25
Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
26
Truth table for ?
p q p ? q T T T T F F F T T F F
T
27
Converse contrapositive
  • For the implication p ? q, the converse is q ? p
  • For the implication p ? q, the contrapositive is
    ?q ? ?p

28
Biconditional
  • A biconditional is a proposition that is true
    when p and q have the same truth values (both
    true or both false)
  • For p and q, the biconditional is denoted as p ?
    q, which can be read as
  • p if and only if q
  • p is necessary and sufficient for q
  • if p then q, and conversely

29
Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
30
Truth table for ?
p q p ? q T T T T F F F T F F F
T
31
Compound propositions
  • Can build compound propositions by combining
    simple propositions using negation and
    connectives
  • Use parentheses to specify order or operations
  • Negation takes precedence over connectives

32
Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of (p ? q)
?? ( p ? r) (r ? q) ? (?p ? r) ?(p ? ?r ) ?
?(p ? r)
33
Logic Bit Operations
  • A bit string is a sequence of 1s and 0s - the
    number of bits in the string is the length of the
    string
  • Bit operations correspond to logical operations
    with 1 representing T and 0 representing F

34
Bit operation examples
Let s1 10011100 s2 11000110 s1 OR s2
11011110 s1 AND s2 10000100 s1 XOR s2
01011010
35
Section 1.1
  • Logic
  • - ends -
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