Logic of Compound Statements

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Logic of Compound Statements

• Introductory Discrete Mathematics (CS/MAT165)

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Compound Statements

• Logical Form and Logical Equivalence
• Conditional Statements
• Valid and Invalid Arguments
• Application Digital Logic Circuits

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Boolean Operations

• Logical
• Combines logical expressions
• Arguments must be logical
• Relational
• Compares logical expressions
• Arguments must be comparable

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Logical Operations

• OR a ? b
• AND a ? b
• NOT a

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Truth Tables
Or
And
Not
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Relational Operations

• Greater than a b
• Less than a
• Equal to a b
• Not greater than a b (also a ? b)
• Not less than a
• Not equal to a ? b

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Logical Form and Logical Equivalence

• Argument
• A sequence of statements (premise) aimed at
  demonstrating the truth of an assertion
  (conclusion).
• Example
• Premise If a fruit is a cherry, then its red.
• Conclusion Therefore, if a fruit is not red,
  then its not a cherry.

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Logical Form and Logical Equivalence

• Premise a ? b (if a then b)
• Conclusion b ? a (If not b then not a)
• or maybe
• Premise a ? b
• Conclusion b ? a
• Is This True?

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Logical Equivalence

• Two expressions are said to be logically
  equivalent when the first is true if and only if
  the second is true
• Examples
• (p) ? p (double negative)
• a ? b ? b ? a (commutative property)
• (a ? b) ? c ? a ? (b ? c) (associative
  property)

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De Morgans Laws

• (p ? q) ? p ? q
• (p ? q) ? p ? q

Augustus De Morgan (1806-1871)
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Tautological Statements Contradictions

• Tautological Statement
• Always true regardless of the values of its
  variables
• Example
• p ? p
• Contradiction
• Always false regardless of the values of its
  variables
• Example
• p ? p

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Study!

• Theorem 1.1.1Logical Equivalencespage 14

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Conditional Statements

• If p then q
• p ? q

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Logical EquivalencesDivision Into Cases

• p ? q ? r ? (p ? r) ? (q ? r)
• Think about it!
• It makes sense!

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Representing If-Then as Or

• p ? q ? p ? q
• How could you prove this?
• Do this!
• Create a truth table for both sides of the
  equivalence

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Negation

• (p ? q) ? p ? q
• Okay, truth tables (again)

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Contrapositive

• p ? q ? q ? p
• Truth tables (one more time)

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Converse / Inverse

• Given
• p ? q
• Converse
• q ? p
• Inverse
• p ? q

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Valid and Invalid Arguments

• Argument
• p ? q ? rq ? p ? r? p ? r (Read therefore p
  implies r)
• This is invalid!
• Why?
• How would you mathematically show its invalid?

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Valid and Invalid Arguments

• p ? q ? r
• q ? p ? r
• ? p ? r
• First, rewrite (i)(p ? q) ? (p ? r)
• Substitute (ii) into (i) (p ? p ? r) ? (p ? r)
• Rewrite (p ? r) ? (p ? r)

• Set this equal to (iii) (p ? r) ? (p ? r) p ?
  r
• Simplify p ? r p ? r
• This is false, therefore the original argument is
  invalid

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Modus Ponens Modus Tollens

• Syllogism
• An argument with two premises and a conclusion
• Modus ponens
• If p then qp ? q
• Modus tollens
• If p then q
• q
• ? p

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Rules of Inference
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Rules of Inference
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ProblemHow to Reason With a Robot

• You might try the ploy successfully used by Mr.
  Spock on the malevolent android 'Norman' (Star
  Trek), wherein he posed the following to Norman
  "Everything I say is a lie. I'm a liar". Norman
  self-fried.
• Can you prove what Spock said is invalid?

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Knights and Knaves

• On a fictional island, all inhabitants are either
  knights, who always tell the truth, or knaves,
  who always lie. The puzzles involve a visitor to
  the island who meets small groups of inhabitants.
  Usually the aim is for the visitor to deduce the
  inhabitants' type from their statements, but some
  puzzles of this type ask for other facts to be
  deduced. The puzzle may also be to determine a
  yes/no question which the visitor can ask in
  order to discover what he needs to know.

Raymond Smullyan (b1919)
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Knights and KnavesQuestion 1

• John and Bill are residents of the island of
  knights and knaves.
• John says We are both knaves.
• Who is who?

Raymond Smullyan (b1919)
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Knights and KnavesQuestion 2

• John If Bill is a knave then I'm a knight.
• Bill We are different.
• Who is who?

Raymond Smullyan (b1919)
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Knights and KnavesQuestion 3

• Logician Are you both knights?
• John answers either Yes or No, but the Logician
  does not have enough information to solve the
  problem.
• Logician Are you both knaves?
• Bill answers either Yes or No, and the Logician
  can now solve the problem.
• Who is who?

Raymond Smullyan (b1919)
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Knights and KnavesQuestion 4

• Here is a rendition of perhaps the most famous of
  this type of puzzle
• John and Bill are standing at a fork in the road.
  You know that one of them is a knight and the
  other a knave, but you don't know which. You also
  know that one road leads to Someplace, and the
  other leads to Nowhere.
• By asking one yes/no question, can you determine
  the road to Someplace?
• By asking one yes/no question, can you determine
  whether John is a knight?
• This version of the puzzle was further
  popularized by a scene in the 1980's fantasy
  film, Labyrinth, in which Sarah (Jennifer
  Connelly) finds herself faced with two doors each
  guided by a two-headed knight. One door leads to
  the castle at the centre of the labyrinth, and
  one to certain doom.

Raymond Smullyan (b1919)
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Digital Logic Circuits

• Time to use the computers!