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Logic
Chapter 3
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Statements and Logical= = Connectives
Section 3.1
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Statements
pStatement:=   A sentence that can be judged either true or false
pNegation of a Statement:  Changes the statement to <= /span>its opposite meaning.  The negation of a true statement must be false and vice-versa.
pExample of Negation
nStatement:  I have ten fingers
nNegation:=   I do not have ten fingers.
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Quantifiers
pWords that change the meaning of a statement (all, none, no, some)
pNegation of statements with quantifiers is more difficult
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Example:  Negate the statement = 220;All dogs have fleas”
pStatement is false, so negation m= ust be true
pNo dogs have fleas?
False statement, cannot be the negation
pNot all dogs have fleas
True statement, this is the negation
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Example:  Negate the statement = 220;No cats drink milk”
pStatement is false, so negation m= ust be true
pAll cats drink milk?
False statement, cannot be the negation
pSome cats drink milk
True statement, this is the negation
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Negations of Statements with = = Quantifiers
All are
Some are not
None are
Some are
Some are
None are
Some are not; Not all are
All are
Form of Negation
Form of Statement
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Types of Statements
pSimple Statement:  Deals with one idea
pCompound Statement:  A statement consisting of = two or more simple statement =
<= span style=3D'font-family:"Times New Roman"'>nConnectives:=   Used to connect the simple statements in a compound statement
(and, or, if → then, if and only if)
nNegated Statement:  Also considered to be a compound = statement
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Symbolism in Logic
pIt is common to represent simple statements by letters (usually p, q, r, and s)
pSaves time and writing when evaluating statements
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Symbolism in Logic
pNegation:=   Symbolized by a tilde ~ and read “not”
pConjunction:=   A compound statement that uses and.
Symbolized by     and read “and”.&= #13;
pDisjunction:=   A compound statement that uses or.
Symbolized by     and read “or”.&#= 13;
nExclusive OR:  One or the other event may occur,= but not both (Eat your supper or go to bed early)
nInclusive OR:  Either event, or both, may occur = (Would you like peas or carrots?)
<= span style=3D'font-family:"Times New Roman"'>nOur disjunctions are understood to be the inclusive OR
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Examples
pLet “My dog has fleas” be represented by p
Let “My cat drinks milk” be represented by q
pWrite the following statements in symbolic form
nMy dog has fleas and my cat drinks milk
np     q
nMy dog does not have fleas or my cat drinks milk
n~p     q
<= span style=3D'font-family:"Times New Roman"'>nMy dog does not have fleas and my cat does not drink milk <= /div>
n~p     ~q
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s68612" DrawAspect= =3D"Content" ObjectID=3D"6"> = <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s68614" DrawAspect= =3D"Content" ObjectID=3D"8"> = <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s68616" DrawAspect= =3D"Content" ObjectID=3D"9">
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Examples
pp:  My dog has fleas
q:  My cat drinks milk
pWrite the following statements in words
np     q
nMy dog has fleas or my cat drinks milk
np     ~q <= /span>
nMy dog has fleas and my cat does not drink milk
n~p     ~q <= /span>
<= span style=3D'font-family:"Times New Roman"'>nMy dog does not have fleas or my cat does not drink milk
n~(p     q) <= /span>
<= span style=3D'font-family:"Times New Roman"'>nIt is not true that my dog has fleas and my cat drinks milk
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s289796" DrawAspec= t=3D"Content" ObjectID=3D"10"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s289797" DrawAspec= t=3D"Content" ObjectID=3D"11"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s289799" DrawAspec= t=3D"Content" ObjectID=3D"13">
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Emphasis in Compound <= span style=3D'position:absolute;top:10.75%;left:5.99%;width:88.2%;height:9.25%'= >= Statements
pA written compound statement with more than one connective often uses a comma to indicate which simple statements are grouped together
pWhen written symbolically, the simple statements on the same side of the comma are grouped within parenthesis
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Examples
pp:  I’ll= go to a dinner
q:  I’ll= go to a movie
r:  I’ll= go to the theater
pWrite the following statements in symbolic form:
nI’ll go to di= nner and a movie, or the theater
n(p     q)     r
pNote:  Since the     is outside the parenthesis and is evaluated last, this is a disjunction
nI’ll go to dinner, and a movie or the theater
np     (q     r) <= /span>
pNote:  Since the     is outside the parenthesis and is evaluated last, this is a conjunction<= /span>
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s101380" DrawAspec= t=3D"Content" ObjectID=3D"21"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s101382" DrawAspec= t=3D"Content" ObjectID=3D"15"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s101388" DrawAspec= t=3D"Content" ObjectID=3D"22">
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Connectives
pConditional:=   Compound statement that uses if…then.  Symbolized by → and read “if…then”.&#= 13;
<= span style=3D'font-family:"Times New Roman"'>nAntecedent:=   Part of the conditional before the →
nConsequent:=   Part of the conditional after the →
pBiconditional:=   Compound statement that uses if and only if.  Symbolized by ↔ and read “if and only if”.<= /i>
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Examples
pp:  The car has gas
q:  The car is running
r:  The car’s speed is 55mph= 3;
pWrite the following statements in words.  Indicate whether each statement is a negation, conjunction, disjunction, conditional, or biconditional.
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Examples (cont.)
p~p → ~q <= /div>
pIf the car is not out of gas then= the car is not running.  <= /span>Conditional
pr → (p     q) <= /span>
pIf the car’s speed is 55mph, then the car has gas and the car is running.  Conditional.
p(~p → ~q)     ~r
pIf the car has no gas then the car is not running, and the car’s speed is not 55mph.  Conjunction.
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s140292" DrawAspec= t=3D"Content" ObjectID=3D"23"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s140294" DrawAspec= t=3D"Content" ObjectID=3D"24">
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Examples (cont.)
pp ↔ q
pThe car has gas if and only if the car is running.  Biconditional. =
p~(q ↔ r) =
pIt is not true that the car is running if and only if the car’s speed is 55mph.  Negation.
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Dominance of Connectives
pDetermines the order in which connectives are evaluated
pSimilar to order of operations in arithmetic
pDoes the statement p     q → r mean
(p     q) → r
or
p     (q → r)?<= /span>
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s158724" DrawAspec= t=3D"Content" ObjectID=3D"25"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s158726" DrawAspec= t=3D"Content" ObjectID=3D"26">
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Dominance of Connectives
Evaluate Last
Most Dominant
Biconditional ↔
Conditional →
Conjunction Disjunction=
Evaluate First
Least Dominant
Negation ~
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R0lGODlhIwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAcABwAO AAwAgQAAAAcGBgsJCgECAwIhjICpYqEPTDwQMHSrRHu5lCkdR2GfJ5apWVnf+BhhxcAFADs= ------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0165.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Examples
pAdd parenthesis to the following statements according to the dominance of connectives.  Identify the type of each statement.
<= span style=3D'font-family:"Times New Roman";visibility:hidden'>p
p~p     q
p(~p)     q Conjunction
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s169988" DrawAspec= t=3D"Content" ObjectID=3D"32"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s169990" DrawAspec= t=3D"Content" ObjectID=3D"33">
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Examples (cont.)
pp     q → r = ;
p(p     q) → r Conditional
<= span style=3D'font-family:"Times New Roman";visibility:hidden'>p
pp ↔ q → r =
pp ↔ (q → r) Biconditional
<= span style=3D'font-family:"Times New Roman";visibility:hidden'>p
pp     q ↔ r     s
p(p     q) ↔ (r     s) Biconditional<= /div>
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Truth Tables for = Negation, Conjunction, and Disjunction
Section 3.2
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0183.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Truth Table
pUsed to determine when a compound statement is true or false
pDepends on whether the underlying simple statements are true or false
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0185.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Truth of a Negation Example
p= Consider the statement:  “I will not end class late”
When do I keep my promise?
pp:  I end class late
pThe statement is ~p
pIf p is true, then ~p is false
I have broken my promise
pIf p is false, then ~p is true
I have kept my promise
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0195.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Negation Truth Table
Two Cases
Case 2
Case 1
~p
p
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Negation Truth Table
Two Cases
T
F
Case 2
F
T
Case 1
~p
p
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Truth of a Conjunction Example
pConsider:  “I will start and end class= on time”
When do I keep my promise?
pp:  I start class on time
q:  I end class on time
pThe statement is p     q
pIf p and q are true then p   =   q is true
I have kept my promise
pOtherwise p     q is false
I have broke my promise
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Conjunction Truth Table
Four Cases
Case 4
Case 3
Case 2
Case 1
p     q
q
p
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Conjunction Truth Table
Four Cases
F
F
F
Case 4
F
T
F
Case 3
F
F
T
Case 2
T
T
T
Case 1
p     q
q
p
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Truth of a Disjunction Example
pConsider:  “I will bring candy or gum = to class”
When do I keep my promise?
pp:  I bring candy to class
q:  I bring gum to class <= /span>
pThe statement is p     q
pIf p and q are false then p     q is false
I have broken my promise
pOtherwise p     q is true
I have kept my promise
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s225284" DrawAspec= t=3D"Content" ObjectID=3D"45"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s225286" DrawAspec= t=3D"Content" ObjectID=3D"46"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s225288" DrawAspec= t=3D"Content" ObjectID=3D"69">
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Disjunction Truth Table
Four Cases
Case 4
Case 3
Case 2
Case 1
p     q
q
p
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Disjunction Truth Table
Four Cases
F
F
F
Case 4
T
T
F
Case 3
T
F
T
Case 2
T
T
T
Case 1
p     q
q
p
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General Procedure for = = Constructing Truth Tables
<= span style=3D'font-family:"Times New Roman"'>1)Determine the type of statement.<= span style=3D'mso-spacerun:yes'>  The answer to the <= span style=3D'position:absolute;top:33.78%;left:5.65%;width:85.4%;height:32.43%= '>truth table will appear under the corresponding symbol.
2)Complete columns under simple statements and negations within parenthesis.
3)Evaluate the connectives within parenthesis.
4)Complete columns under remaining simple statements and negations.
5)Evaluate the remaining connectives.
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Example:  Construct a truth table for ~(p     ~q)
F
F
T
F
F
T
T
T
~q)
(p
~
q
p
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Example:  Construct a truth table for ~(p     ~q)
2nd
3rd
1st
4th
=
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T
T
F
F
F
F
F
F
F
T
T
F
T
T
T
F
F
T
F
T
T
F
T
T
~q)
(p
~
q
p
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Example (cont.)
pNote:=   The last-completed column of a truth table contains the “answers”
pIf p is true and q is false, what is the truth value of ~(p     ~q)?
pFalse
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Example:  Construct a truth table for (p     ~q)     ~p
~p
F
F
T
F
F
T
T
T
~q)
(p
q
p
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Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0415_image079.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLgAeAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAkACAAU ABAAgwAAAAAAAAsJCgUEBAoICQgHBwMDAwkICAwKCwcGBgcGBwECAwECAwECAwECAwECAwRNEMgp RQiC6g3GHVxIBEUREOFmGNKaUkdwSPH8IgE4echropNRIWVJbBKYEDKjKXJGipDitOGFcLpJ7bWd uF6AL+AHBggB2LKEt1QDihEAOx== ------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0425.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Example:  Construct a truth table for (p     ~q)     ~p
4th
T
T
F
F
~p
5th
2nd
3rd
1st
=
T
T
F
F
F
F
T
F
F
F
T
F
T
T
T
T
F
T
F
F
F
T
T
T
~q)
(p
q
p
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Example (cont.)
pIf p is true and q is true, what = is the truth value of
(p     ~q)     ~p? =
pFalse
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s351236" DrawAspec= t=3D"Content" ObjectID=3D"58"> <= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s351238" DrawAspec= t=3D"Content" ObjectID=3D"59">
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0268_image085.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEgARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAQABQAK AAkAggAAAAsJCgcGBggHBwoICQwKCwUEBAkHCAMgCBoioRAMQkYExShT4iEKcUQaVAKiBwJclKHW NTkXwCQAOy== ------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0268_image086.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEwARAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAUABQAK AAkAggAAAAcGBgsJCgoICQgHBwUEBAwKCwkHCAMgGALcEoO0RgYoZprCxpmHxXANyYTd12xjNlXR xDxKkwAAOw== ------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0269.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Truth Value
without a Truth Table<= /div>
p= Sometimes we wish to determine the truth value of a compound statement based on the truth values of each individual simple statement <= /span>
pWe may do this without constructing the entire truth table
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0270.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Example
pDetermine the truth value of ~p     q
if p is true and q is true
p
p~p     q
pF      T
pT
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Example
pDetermine the truth value of ~(p<= span style=3D'mso-spacerun:yes'>     ~q)
if p is true and q is false
<= span style=3D'font-family:"Times New Roman";visibility:hidden'>p
p~(p     ~q) =
p~(T     T) <= /div>
p~(T)
pF
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Example
pConstruct a truth table for the statement:
“The dog is sleeping, or it is barking and not obeying orders” <= /div>
np:  The dog is sleeping
q:  The dog is barking
r:  The dog is obeying orders
n <= /div>
nThe statement is p     (q     = ~r)
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~r)
(q
p
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Logic
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file:///C:/6E4BB24E/chapter_3_presentation_files/slide0293.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Example (cont.)
pIf the dog is sleeping, barking a= nd obeying orders, what is the truth value of the statement?
pTrue
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0294.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Example
pDetermine the truth value of the following statement:
“Dogs meow or cats do not bark” pp:  Dogs meow False
q:  Cats bark False
p
pp     ~q <= /div>
pF     T
pT
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Truth Tables for = Conditional and Biconditional
Section 3.3
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0301.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Truth of a Conditional Example
pConsider:  “If you are good, Santa will bring presents”
When does Santa keep his promise?
pp:  You are good q:  S= anta brings presents
pThe statement is p → q
pIf p is true and q is false
Santa has broken his promise
pOtherwise
Santa has kept his promise
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0416.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Conditional Truth Table
Four Cases
F
F
Case 4
T
F
Case 3
F
T
Case 2
T
T
Case 1
p <= b>→ q
q
p
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Conditional Truth Table
Four Cases
T
F
F
Case 4
T
T
F
Case 3
F
F
T
Case 2
T
T
T
Case 1
p <= b>→ q
q
p
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Truth of a Biconditional Example
pConsider:  “You get an A in class if a= nd only if you deserve it”
What does this mean?
pIf you get an A then you deserve it AND if you deserve it then you get an A =
pp:  You get an A q:  Y= ou deserve it
pThe statement is p ↔ q
 p ↔ q means (p → q)    (q → p)<= /span>
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s397316" DrawAspec= t=3D"Content" ObjectID=3D"134">
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Example:  Construct a truth table for= (p → q)    (q &#= 8594; p)
p)
→
(q
F
F
T
F
F
T
T
T
q)
→
(p
q
p
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Example:  Construct a truth table for= (p → q)    (q &#= 8594; p)
4th
F
F
T
T
p)
6th
T
F
T
T
→
3rd
F
T
F
T
(q
7th
2nd
5th
1st
<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
<= span style=3D'font-size:133%;mso-special-format:lastCR;display:none'>
T
F
T
F
F
F
F
T
T
F
T
F
F
F
F
T
F
T
T
T
T
T
T
T
=
q)
→
(p
q
p
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image/gif R0lGODlhLgAfAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAkACQAU ABAAgwAAAAAAAAsJCgUEBAoICQgHBwMDAwkICAwKCwcGBgcGBwECAwECAwECAwECAwECAwRNEMgp RQiC6g3GHVxIBEUREOFmGNKaUkdwSPH8IgE4echropNRIWVJbBKYEDKjKXJGipDitOGFcLpJ7bWd uF6AL+AHBggB2LKEt1QDihEAOx== ------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0418.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Biconditional Truth Table
Four Cases
F
F
Case 4
T
F
Case 3
F
T
Case 2
T
T
Case 1
p ↔ q
q
p
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Biconditional Truth Table
Four Cases
T
F
F
Case 4
F
T
F
Case 3
F
F
T
Case 2
T
T
T
Case 1
p ↔ q
q
p
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Self-Contradictions, Tautologies = and Implications
pSelf-Contradiction:= =   A compound statement that is always false
pTautology:=   A compound statement that is always true
pImplication:=   A conditional statement that is a tautology
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Example:  Construct a truth table for (p     q)     (~p     ~q)
~q)
(~p
F
F
T
F
F
T
T
T
q)
(p
q
p
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Example:  Construct a truth table for (p     q)     (~p     ~q)
4th
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6th
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3rd
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7th
2nd
5th
1st
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F
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T
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Example (cont.)
pIs this a self-contradiction, tautology, and/or implication?
pSelf-Contradiction<= /div>
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Example
pConstruct a truth table for (p     q) → (~q →r)
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file:///C:/6E4BB24E/chapter_3_presentation_files/slide0431_image140.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLgAfAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAkACQAU ABAAgwAAAAAAAAsJCgUEBAoICQgHBwMDAwkICAwKCwcGBgcGBwECAwECAwECAwECAwECAwRNEMgp RQiC6g3GHVxIBEUREOFmGNKaUkdwSPH8IgE4echropNRIWVJbBKYEDKjKXJGipDitOGFcLpJ7bWd uF6AL+AHBggB2LKEt1QDihEAOx== ------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0354.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Example (cont.)
pIs this a self-contradiction, tautology, and/or implication?
pTautology and Implication<= /span>
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0355.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Example
pDetermine the truth value of ~(p → q)     p
if p is false and q is false
p~(p → q)     p
p~(F → F)     F
p~T     F
pF     F
pF
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Example
pDetermine the truth value of the following statement:
“If fish swim then fish do not fly”
pp:  Fish swim True
pq:  Fish fly False
pp → ~q
pT → T
pT
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0367.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Equivalent Statements
Section 3.4
------=_NextPart_01CA8E06.2D850240 Content-Location: file:///C:/6E4BB24E/chapter_3_presentation_files/slide0368.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Chapter 3: Logic
Equivalent Statements
p Two statements that have exactly t= he same truth values in the answer columns of their truth tables
pSymbolized
<= o:OLEObject Type=3D"Embed" ProgID=3D"Equation.3" ShapeID=3D"_x0000_s453636" DrawAspec= t=3D"Content" ObjectID=3D"538">
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Example:  Are ~(p     q) and
~p     ~q equivalent statements?
F
F
T
F
F
T
T
T
~q
~p
q)
=
(p
~
q
p
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Example:  Are ~(p     q) and
~p     ~q equivalent statements?
2nd
3rd
1st
2nd
3rd
1st
4th
T
T
T
F
F
F
T
F
F
F
T
T
T
F
F
T
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
T
T
F
T
T
~q
~p
q)
(p
~
q
p
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