The Logic of Compound Statements

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The Logic of Compound Statements

• Prof. Milon
• Coll 147

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Section 1.1

• Logical Form and Logical Equivalence

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Logic Lecture
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Statements

• A statement is a sentence that is either true or
  false, but not both.
• Statements
• It is raining.
• I am carrying an umbrella.
• Not statements
• Hello.
• Are you there?

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Logical Operators

• Binary operators
• Conjunction and.
• Disjunction or.
• Unary operator
• Negation not.
• Other operators
• XOR exclusive or
• NAND not both
• NOR neither

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Logical Symbols

• Statements are represented by letters p, q, r,
  etc.
• ? means and.
• ? means or.
• ? means not.

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Examples

• Basic statements
• p It is raining.
• q I am carrying an umbrella.
• Compound statements
• p ? q It is raining and I am carrying an
  umbrella.
• p ? q It is raining or I am carrying an
  umbrella.
• ?p It is not raining.

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False Negations

• Statement
• Everyone likes me.
• False negation
• Everyone does not like me.
• True negation
• Someone does not like me.

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False Negations

• Statement
• Someone likes me.
• False negation
• Someone does not like me.
• True negation
• No one likes me.

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Truth Table of an Expression

• Make a column for every variable.
• List every possible combination of truth values
  of the variables.
• Make one more column for the expression.
• Write the truth value of the expression for each
  combination of truth values of the variables.

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Truth Table for and

• p ? q is true if p is true and q is true.
• p ? q is false if p is false or q is false.

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Truth Table for or

• p ? q is true if p is true or q is true.
• p ? q is false if p is false and q is false.

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Truth Table for not

• ?p is true if p is false.
• ?p is false if p is true.

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Example Truth Table

• Truth table for the statement (?p) ? (q ? r ).

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Logical Equivalence

• Two statements are logically equivalent if they
  have the same truth values for all combinations
  of truth values of their variables.

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Example Logical Equivalence

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

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DeMorgans Laws

• ?(p ? q) ? (?p) ? (?q)
• ?(p ? q) ? (?p) ? (?q)
• (C) If it is not true that
• i lt size value ! arrayi
• then it is true that
• i gt size value arrayi
• If it is not true that x ? 5 or x ? 10, then it
  is true that x gt 5 and x lt 10.

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Tautologies and Contradictions

• A tautology is a statement that is logically
  equivalent to T.
• A contradiction is a statement that is logically
  equivalent to F.
• Some tautologies
• p ? ?p
• p ? ?q ? (? p ? q)
• Some contradictions
• p ? ?p
• p ? q ? (?p ? ?q)

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Section 1.2

• Conditional Statements

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The Conditional

• A conditional statement is a statement of the
  form
• p ? q
• p is the hypothesis.
• q is the conclusion.
• Read p ? q as p implies q.

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Truth Table for the Conditional

• p ? q is true if p is false or q is true.
• p ? q is false if p is true and q is false.

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Example Conditional Statements

• If it is raining, then I am carrying an
  umbrella.
• This statement is true
• when I am carrying an umbrella (whether or not it
  is raining), and
• when it is not raining (whether or not I am
  carrying an umbrella).

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The Contrapositive

• The contrapositive of p ? q is ?q ? ?p.
• The statements p ? q and ?q ? ?p are logically
  equivalent.

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The Converse and the Inverse

• The converse of p ? q is q ? p.
• The inverse of p ? q is ?p ? ?q.

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Is this logical?
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The Biconditional

• The statement p ? q is the biconditional of p and
  q.
• p ? q is logically equivalent to
• (p ? q) ? (q ? p).

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Other Logical Operators
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Exclusive-Or

• The statement p ? q is the exclusive-or of p and
  q.
• p ? q is defined by

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Exclusive-Or

• p ? q means one or the other, but not both.
• p ? q is logically equivalent to
• (p ? q) ? ?(q ? p)
• p ? q is also logically equivalent to
• ?(p ? q)
• p ? q is also logically equivalent to
• (p ? ?q) ? (q ? ?p)

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Section 1.3

• Valid and Invalid Arguments

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Arguments

• An argument is a sequence of statements.
• The last statement is the conclusion.
• All the other statements are the premises.
• A mathematical proof is an argument.

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Validity of an Argument

• An argument is valid if its conclusion is true
  when its premises are true.
• Otherwise, the argument is invalid.
• An invalid argument is called a fallacy.

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Example Invalid Arguments with True Conclusions

• An argument may be invalid even though its
  conclusion is true.
• If I eat my vegetables, Ill be big and strong.
• Im big and strong.
• Therefore, I ate my vegetables.
• A true conclusion does not ensure that the
  argument is valid.

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Example Valid Arguments with False Conclusions

• An argument may be valid even though its
  conclusion is false.
• If I say hmmm a lot, then people will think Im
  smart.
• I say hmmm a lot.
• Therefore, people think Im smart.
• A false conclusion does not ensure that the
  argument is invalid.

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The Form of an Argument

• The validity of an argument can be determined by
  its form rather than by the truth of its
  conclusion.
• Let the premises be P1, , Pn.
• Let the conclusion be C.
• The argument is valid if
• P1 ? ? ? Pn ? C
• is a tautology.

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Example

• I will ride my bike today.
• If it is windy and I ride my bike, then I will
  get tired.
• It is windy.
• Therefore, I will get tired.

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Example

• p I will ride my bike today.
• q It is windy.
• r I will get tired.
• Argument
• p
• q ? p ? r
• q
• ? r

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Example
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Modus Ponens

• Modus ponens is the argument form
• p ? q
• p
• ? q
• This is also called a direct argument.

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Examples of Modus Ponens

• If it is raining, then I am carrying my umbrella.
  It is raining. Therefore, I am carrying my
  umbrella.
• If pigs can fly, then I am carrying my umbrella.
  Pigs can fly. Therefore, I am carrying my
  umbrella.

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Modus Tollens

• Modus tollens is the argument form
• p ? q
• ?q
• ? ?p
• This is also called an indirect argument.

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Examples of Modus Tollens

• If it is raining, then I am carrying my umbrella.
  I am not carrying my umbrella. Therefore, it is
  not raining.
• If pigs can fly, then I am carrying my umbrella.
  I am not carrying my umbrella. Therefore, pigs
  cannot fly.

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Fallacies

• A fallacy is an invalid argument form.
• Two common fallacies
• The fallacy of the converse.
• The fallacy of the inverse.

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The Fallacy of the Converse

• The fallacy of the converse is the invalid
  argument form
• p ? q
• q
• ? p
• This is also called the fallacy of affirming the
  consequent.

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Example

• If it is raining, then I am carrying an umbrella.
  I am carrying an umbrella. Therefore, it is
  raining.

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Fallacy of the Inverse

• The fallacy of the inverse is the invalid
  argument form
• p ? q
• ?p
• ? ?q
• This is also called the fallacy of denying the
  antecedent.

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Example

• If pigs can fly, then I am carrying an umbrella.
  Pigs cannot fly. Therefore, I am not carrying an
  umbrella.

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Section 1.4

• Application Digital Logic Circuits

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Logic Gates

• Three basic logic gates
• AND-gate
• OR-gate
• NOT-gate
• Two other gates
• NAND-gate (NOT-AND)
• NOR-gate (NOT-OR)

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AND-Gate

• Output is 1 if both inputs are 1.
• Output is 0 if either input is 0.

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OR-Gate

• Output is 1 if either input is 1.
• Output is 0 if both inputs are 0.

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NOT-Gate

• Output is 1 if input is 0.
• Output is 0 if input is 1.

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NAND-Gate

• Output is 1 if either input is 0.
• Output is 0 if both inputs are 1.

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NOR-Gate

• Output is 1 if both inputs are 0.
• Output is 0 if either input is 1.

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Disjunctive Normal Form

• A logical expression is in disjunctive normal
  form if
• It is a disjunction of clauses.
• Each clause is a conjunction of variables and
  their negations.
• Each variable or its negation appears in each
  clause exactly once.

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Examples Disjunctive Normal Form

• p ? q ? (p ? q) ? (?p ? q) ? (?p ? ?q).
• p ? q ? (p ? q) ? (?p ? ?q).
• p q ? (p ? ?q) ? (?p ? q) ? (?p ? ?q).
• p ? q ? ?p ? ?q.