Section 1-4 Logic

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Section 1-4 Logic

• Katelyn Donovan
• MAT 202
• Dr. Marinas
• January 19, 2006

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What is a Statement?

• A statement is a sentence that is either true or
  false, but not both.
• Which of the following are statements?
• Lauren has blue eyes.
• Bush is the best president.
• He smells.

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How did you do?

• 1 is a statement since Lauren was identified as
  the person with blue eyes.
• 2 is not a statement because it can be true and
  false, depending on who you ask.
• 3 is not a statement because the he who smells
  is not identified.

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What is a Negation?

• The negation of a statement is a statement with
  the opposite truth value of the given statement.
• If a statement is true, the negation is false and
  if a statement is false, the negation is true.
• Ex. Statement It is raining now.
• Negation It is not raining now.

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Truth Tables

• Truth tables are used to show all possible
  True-False patterns for statements.
• The symbol p represents a statement and the
  symbol p (read as not p) represents a
  negation.
• The Truth Table for p and p is shown below.

Statement p Negation p
T F
F T
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Compound Statements

• Compound Statements are two statements with a
  connector such as and/or.
• The symbol represents and.
• The symbol v represents or.

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Conjunction

• A conjunction is a compound statement formed by
  joining two statements with the connector AND. 
  The conjunction "p and q" is symbolized by p
  q.  A conjunction is true when both of its
  combined parts are true otherwise it is false.
• The Truth Table for Conjunction is shown below

p q p q
T T T
T F F
F T F
F F F
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Disjunction

• A disjunction is a compound statement formed by
  joining two statements with the connector OR. The
  disjunction
• p or q is symbolized by p V q. The disjunction
  is false only when both p and q are false,
  everywhere else its true.
• The Truth Table for Disjunction is shown below

p q p v q
T T T
T F T
F T T
F F F
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Conditionals

• Conditionals are statements written as
• if p, then q or p ? q
• Conditionals are also known as implications
• The statement after the if is the hypothesis
  and the statement after the then is the
  conclusion.
• The Truth Table for conditional (implication) is
  below

p q p ? q
T T T
T F F
F T T
F F T
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Example of a Conditional

• IF Ariel kisses Eric, THEN she will remain human.
  
• p q Conditional
• T T Ariel kisses Eric she will remain
  T human.
• T F Ariel kisses Eric she does not remain
  F human.
• F T Ariel does not kiss Eric she remains
  human. T
• F F Ariel does not kiss Eric she does not
  remain T human.

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Conditionals are written in many ways

• If p, then q
• If p, q
• q, if p
• p implies q
• p only if q
• p is sufficient condition for q
• q is a necessary condition for p

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Any Implication p ? q has three related
implication statements

• Statement If p, then q. p
  ? q
• Converse If q, then p.
  q ? p
• Inverse If not p, then not q. p ?
  q
• Contrapositive If not q, then not p. q ? p

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Example

• Statement If I am in Miami, then I am in
  Florida. (p ? q)
• Converse If I am in Florida, then I am in Miami.
  (q ? p)
• Inverse If I am not in Miami, then I am not in
  Florida. (p ? q)
• Contrapositive If I am not in Florida, then I am
  not in Miami. (q ? p)

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PROPERTY TIME!

• Equivalence of a statement and its
  contrapositive.
• The implication p ? q and its contrapositive q ?
  p are logically equivalent.

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Biconditional Statements

• Connecting a statement and its converse with the
  connective and gives (p ? q) (q ? p )
• This compound statement can be written as p 1 q
  and is read as p if and only if q
• This statement is called a biconditional.

p q p 1 q
T T T
T F F
F T F
F F T
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• HAVE FUN AND GOOD LUCK!
• DO NOT FORGET TO COME SEE ME
• IN THE MATH LAB
• FOR ADDITIONAL ASSISTANCE ?