Transcript Slide 1

Discrete Structures
Chapter 2: The Logic of Compound Statements
2.1 Logical Forms and Equivalence
Logic is a science of the necessary laws of thought, without which no
employment of the understanding and the reason takes place.
– Immanuel Kant, 1724 – 1804
Foundation for the Metaphysics of Morals, 1785
2.1 Logical Forms and Equivalences
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Logic
• Logic is the study of reasoning; specifically
whether reasoning is correct.
Note: We use p, q, and r to represent propositions.
• Logic does:
– Assess if an argument is valid/invalid
• Logic does not directly:
– Assess the truth of atomic statements
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Statements
• Statement or Proposition – any sentence that
is true or false but not both
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Statements
• Examples of Statements:
1. George Washington was the first president of the
United States.
2. Baltimore is the capital of Maryland.
3. Seventeen is an even number.
• The above statements are either true (1) or
false (2, 3)
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Not Statements
• These are not statements:
1. Earth is the only planet in the universe that
contains life.
2. Buy two tickets to the rock concert for Friday
3. Why should we study logic?
• None of the above can be determined to be
true or false so they are not statements.
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Ambiguity
• It is possible that a sentence is a statement yet we can not determine
its truth or falsity because of an ambiguity or lack of qualification.
• Examples:
1.
2.
3.
Yesterday it was cold.
He thinks Philadelphia is a wonderful city.
Lucille is a brunette.
For (1) we need to determine what we mean by the word “cold”
For (2) we need to know whose opinion is being considered.
For (3) it depends on which Lucille we are discussing.
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Compound Statements
• In speech and writing, we combine
propositions using connectives such as and or
even or.
• For example, “It is snowing” and “It is cold”
can be combined into a single proposition “it is
snowing and it is cold.”
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Compound Statements
• Let:
p = “It is snowing.”
q = “It is cold.”
Connective
Symbol
Name
Not
~





Negation
And
Or (inclusive)
Or (exclusive)
If…then…
If and only if (iff)
Example
~q: It is not the case that it is
cold.
Conjunction
pq: It is snowing and it is cold.
Disjunction
pq: It is snowing or it is cold.
Exclusive Or
Conditional
Biconditional
2.1 Logical Forms and Equivalences
pq: It is snowing or it is cold but
not both.
pq: If it is snowing, then it is
cold.
pq: It is snowing iff it is cold.
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Translating: English to Symbols
• The word but translates the same as and.
“Jim is tall but he is not heavy” translates to
“Jim is tall AND he is not heavy”
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Translating: English to Symbols
• The words neither-nor translates the same as
not.
“Neither a borrower nor a lender be” translates to
“Do NOT be a borrower and do NOT be a
lender”
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Truth Values
• If sentences are statements then the must have
well-defined truth values meaning the
sentences must be either true or false.
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Negation
• Definition
If p is a statement variable, the negation of p
is “not p” or “it is not the case that p” and is
denoted p. It has the opposite truth value
from p:
if p is true, p is false
if p is false, p is true
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Negation Truth Table
The truth value for
negation are
summarized in the
truth table on the
right.
p
p
T
F
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Conjunction
• Definition
If p and q are statement variables, the
conjunction of p and q is “p and q”, denoted
p  q. It is true when BOTH p and q are true.
If either p or q is false, or both are false, p  q
is false.
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Conjunction Truth Table
• The truth value for
conjunction are
summarized in the
truth table on the
right.
p
q
T
T
T
F
F
T
F
F
2.1 Logical Forms and Equivalences
p q
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Disjunction (Inclusive Or)
• Definition
– If p and q are statement variables, the disjunction
of p and q is “p or q”, denoted p  q. It is true
when either p is true, q is true, or both p and q are
true. If both p and q is false, p  q is false.
• Example
– You may have cream or sugar with your coffee
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Disjunction Truth Table
• The truth value for
conjunction are
summarized in the
truth table on the
right.
p
q
T
T
T
F
F
T
F
F
2.1 Logical Forms and Equivalences
p q
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Exclusive Or
• Definition
– If p and q are statement variables, the exclusive or
of p and q is “p or q”, denoted pq. It is true
when either p is true or when q is true, but not
both. If both p and q is false, p  q is false. If
both p and q is true, p  q is false.
• Example
– Your meal comes with soup or salad.
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Exclusive Or Truth Table
• The truth value for
conjunction are
summarized in the
truth table on the
right.
p
q
T
T
T
F
F
T
F
F
2.1 Logical Forms and Equivalences
pq
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Statement Form
• Definition
– A statement form is an expression made up of
statement variables such as p, q, and r and logical
connectives that becomes a statement when actual
statements are substituted for the component
statement variables.
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Example
• Construct a truth table for the statement form
(pq)(pq).
p
q
T
T
T
F
F
T
F
F
p
q
pq
pq
2.1 Logical Forms and Equivalences
(pq)(pq)
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Tautology
• Definition
– A tautology is a statement form that is always true
regardless of the truth values of the individual
statements substituted for its statement variables.
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Contradiction
• Definition
– A contradiction is a statement form that is always
false regardless of the truth values of the individual
statements substituted for its statement variables.
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Logically Equivalent
• Definitions
– Two statement forms are logically equivalent iff
they have identical truth values for each possible
substitution of statements for their statement
variables. Logical equivalence of statement forms
P and Q is denoted by PQ.
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Testing for Logical Equivalence
1. Construct a truth table with one column for
the truth values of P and another column for
the truth values of Q.
2. Check the combination of truth values of the
statement variables.
a. If in each row the truth value of P is the same as
the truth value of Q, then PQ.
b. If in each row the truth value of P is not the same
as the truth value of Q, then PQ.
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Example
Are the statement forms (pq) and pq
logically equivalent?
p
q
T
T
T
F
F
T
F
F
p
q
pq
(pq)
(pq)
 (pq)  pq
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Theorem 2.1.1 – Logical
Equivalence
• Given any statement variables p, q, and r, a tautology t, and a
contradiction c, the following logical equivalences hold.
1. C om m utative L aw s:
pq q p
pq q p
2. A ssociative L aw s:
 p  q r
 p  q r
 p  q  p
 p  q  p 
3. D istributive L aw s:
p  q  r    p  q   p  r 
4. Identity L aw s:
pt p
pc p
5. N egation L aw s:
p ~ p  t
p ~ p  c
6. D ouble N e gative L aw :
~ ~ p   p
7. Idem potent L aw s:
p p  p
p p  p
8. U niversal B ound L aw s:
ptt
pcc
9. D e M organ's L aw s:
~
10. A bsorption L aw s:
p   p  q  p
p   p  q  p
11. N egations of t and c:
~t  c
~ct
 p  q  ~
p ~ q
2.1 Logical Forms and Equivalences
p  q  r    p  q   p  r 
~  p  q  ~ p ~ q
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De Morgan’s Laws
• The negation of an and statement is logically
equivalent to the or statement in which each
component is negated.
(p  q)  p  q
• The negation of the or statement is logically
equivalent to the and statement in which each
component is negated.
(p  q)  p  q
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Example – pg. 38 # 33
• Use De Morgan’s Laws to write negations for
the statement.
-10 < x < 2
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Example – pg. 38 # 51
• Use Theorem 2.1.1 to verify the logical
equivalences. Supply a reason for each step.
p  (q  p)  p
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Next time
• Continue with logical equivalence.
• Discuss valid and invalid arguments.
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