Statements and Logical Connectives

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Section 3.1

• Statements and Logical Connectives

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Logic

• Use of deductive reasoning to come to a
  conclusion given a set of information
• Founders Aristotle, Leibniz, Boole

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Statements

• Foundation of logic is the statement.
• Examples of statements
• Sony is the largest employer in Japan.
• It is snowing.
• Professor Ryan teaches math.
• A statement is a sentence whose truth can be
  determined.

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Logical Connectives

• And, Or, Ifthen
• And
• You must feed the meter and you must park
  between the lines.
• Both
• Or
• You must feed the meter or you must park between
  the lines.
• One or the other or both (INCLUSIVE OR)

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Logical Connectives

• If then
• If the parking police see your expired meter,
  then you will get a ticket.
• Relates two ideas

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Symbols for Statements

• Lower case letters represent statements
• p You must feed the meter.
• q You must park between the lines.

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Symbols for Connectives

• Symbols represent logical connectives
• ? AND
• V OR
• NOT
• ? IFTHEN

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And Conjunction

• Example
• You must feed the meter AND you must park between
  the lines.
• Let
• p You must feed the meter.
• q You must park between the lines.
• ? AND
• p ? q
• CONJUNCTION

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Or Disjunction

• Example
• You must feed the meter OR you must park between
  the lines.
• Let
• p You must feed the meter.
• q You must park between the lines.
• V OR
• p V q
• DISJUNCTION

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Not Negation

• Example
• You must not feed the meter.
• Let
• p You must feed the meter.
• NOT
• p
• NEGATION

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If then Conditional

• Ifthen symbolized by ?
• p ? q is read If p, then q
• If you are over 16, then you may drive.
• p You are over 16.
• q You may drive.
• p ? q

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Translate English into Symbols

• Example
• Sony will replace the product and Sony will not
  give a credit toward a future purchase.
• Let
• p Sony will replace the product.
• q Sony will give a credit toward a future
  purchase
• ? AND
• p ? q

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Translate English into Symbols

• Example
• Sony will not replace the product and Sony will
  give a credit toward a future purchase.
• Let
• p Sony will replace the product.
• q Sony will give a credit toward a future
  purchase
• ? AND
• p ? q

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Other Words for And

• Consider the following
• Sony will not replace the product,
• but Sony will give a credit toward a future
  purchase.
• however, Sony will give a credit toward a future
  purchase.
• nevertheless, Sony will give a credit toward a
  future purchase.
• What is the sense of each of these statements?
• All the blue words mean and.

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Negation and Parentheses It is not the case
that

• p It is warm.
• q It is sunny.
• NOT
• ? AND
• p ? q
• It is not warm and it is sunny.
• What would (p ? q) be?
• It is not the case that it is warm and it is
  sunny.

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Parentheses and the Conditional Statement

• p You are over 16.
• q You may drive.
• What does p ? q mean?
• If you are not over 16 then you may not drive.
• What does (p ?q) mean?
• It is not the case that if you are over 16
  then you may drive.

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Order of Operations

• p V q gt r
• Which operation is done first?
• Order of Operations ( )
• ? V
• ?