Logical Form and Logical Equivalence

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Logical Form and Logical Equivalence

• M260 2.1

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Logical Form Example 1

• If the syntax is faultyor execution results in
  division by zero,then the program will generate
  an error message.
• Thereforeif the computer does not generate an
  error messagethen the syntax is correctand the
  execution does not result in division by zero.

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Logical Form Example 2

• If x is a Real number such that xlt-2 or xgt2,then
  x2gt4.
• Thereforeif x2?4,then x?-2 and x?2.

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Logical Form Example 1

• If (the syntax is faulty)or (execution results
  in division by zero),then (the program will
  generate an error message).
• Thereforeif (the computer does not generate an
  error message)then (the syntax is correct)and
  (the execution does not result in division by
  zero).

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Logical Form Example 1

• If (p)or (q),then (r).
• Thereforeif (not r)then (not p)and (not q).

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Logical Form Example 2

• If (xlt-2) or (xgt2),then (x2gt4).
• Thereforeif (x2?4),then (x?-2) and (x?2).

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Logical Form Example 2

• If (p) or (q),then (r).
• Thereforeif (not r),then (not p) and (not q).

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Logical Form vs Content

• Examples 1 and 2 have the same formIf p or q,
  then r.therefore if not r, then not p and not q.
• These examples have different values for the
  propositional variables p and q.

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Formal Logic Goals

• Avoid Ambiguity
• Obtain Consistency
• Elucidate Proof Mechanisms

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Mathematical Vocabulary

• New terms are defined using previously defined
  terms.
• Initial terms remain undefined.
• Undefined terms in logic sentence, true, false.

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Logic Symbols ? ?

• denotes not
• Negation of p is p.

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Logic Symbols ? ?

• ? denotes and
• Conjunction of p and q is p ? q.
• ? denotes or
• Disjunction of p and q is p ? q.
• Precedence first then ? and ? (unordered)

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Truth Values

• True
• False

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Precedence Examples

• p ? q
• p ? q
• (p ? q)

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Let p, q and r be 0ltx, xlt3, and x3

• Rewrite x ?3
• q ? r
• Rewrite 0ltxlt3
• p?q
• Rewrite 0ltx?3
• p?(q ? r)

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Negation Truth Table
p p
T F
F T
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Conjunction Truth Table
p q p?q
T T T
T F F
F T F
F F F
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Disjunction Truth Table
p q p ? q
T T T
T F T
F T T
F F F
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Statement Form

• Statement variables
• Logical connectives
• Truth table

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Exclusive Or

• p or q but not both
• (p ? q) ? (p ? q)
• Do a truth table

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Exclusive Or Truth Table
p q p ? q p ? q (p ? q) (p ? q) ? (p ? q)




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Exclusive Or Truth Table
p q p ? q p ? q (p ? q) (p ? q) ? (p ? q)
T T
T F
F T
F F
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Exclusive Or Truth Table
p q p ? q p ? q (p ? q) (p ? q) ? (p ? q)
T T T T F
T F T F T
F T T F T
F F F F T
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Exclusive Or Truth Table
p q p ? q p ? q (p ? q) (p ? q) ? (p ? q)
T T T T F F
T F T F T T
F T T F T T
F F F F T F
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Logical Equivalence

• Statement Forms are logically equivalent if, and
  only if, they have the same truth tables.
• P ? Q

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Logical Equivalence Examples

• 6gt2 2lt6
• p ? q q ? p
• p (p)

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De Morgans Laws

• (p ? q) ? p ? q
• (p ? q) ? p ? q
• Do truth tables

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(p ? q) ? p ? q
p q p q p ? q (p ? q) p ? q




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(p ? q) ? p ? q
p q p q p ? q (p ? q) p ? q
T T
T F
F T
F F
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(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 T F F
F T T F T F F
F F T T F T T
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Practice Negations

• John is six feet tall and weighs at least 200
  pounds.
• John is not six feet tall or he weighs less than
  200 pounds.

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Practice Negations

• The bus was late or Toms watch was slow.
• The bus was not late and Toms watch was not
  slow.

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Jim is tall and thin.

• Logical And and Or are only allowed between
  statements.

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Tautologies and Contradictions

• A tautology is a statement form that is always
  true regardless of the values of the statement
  variables.
• A contradiction is a statement form that is
  always false regardless of the values of the
  statement variables

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Logically Equivalent Forms

• Commutative laws
• Associative laws
• Distributive laws
• Identity laws
• Negation laws
• Double negative law

• Idempotent laws
• De Morgans laws
• Universal bound laws
• Absorption laws
• Negations of tautologies and contradictions

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Logical Equivalences

• p?q ?_________ p?q ? ________
• (p?q)?r ?_______ (p?q)?r ?_______ 
• p?(q?r) ? ______ p?(q?r) ? _______
• p?t ?__________ p?c ? __________
• p?p ? _________ p?p ? _________
• (p) ? ________
• p?p ? __________ p?p ? __________
• (p?q ) ? _______ (p?q ) ? _______
• p?t ? __________ p?c ? __________
• p?(p?q) ? ______ p?(p?q) ? ______
• t ? ___________ c ? ___________

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Logical Equivalences

• p?q ? q?p p?q ? q?p
• (p?q)?r ? p?(q?r) (p?q)?r ? p?(q?r)  
• p?(q?r) ? (p?q)? (p? r)
• p?(q?r) ? (p?q)? (p? r)
• p?t ?p p?c ? p
• p?p ? t p?p ? c
• (p) ? p
• p?p ? p p?p ? p
• (p?q ) ? p?q (p?q ) ? p?q
• p?t ? t p?c ? c
• p?(p?q) ? p p?(p?q) ? p
• t ? c c ? t