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{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "Module 7 : Discrete Mathematics" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 3 "" 0 "" {TEXT -1 26 "701 : Logic & Truth Tables" }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 17 "O B J E C T I V E" }}
{PARA 0 "" 0 "" {TEXT -1 135 "We're going to use Maple to create truth
 tables for logical expressions. To do this we are going to define som
e custom built functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 9 "S E T U P" }}
{PARA 0 "" 0 "" {TEXT -1 252 "In this project we will use the followin
g command packages. Type and execute this line before begining the pro
ject below. If you re-enter the worksheet for this project, be sure to
 re-execute this statement before jumping to any point in the workshee
t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart;; with(plots):" }}
{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been r
edefined\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 
"_____________________________________________________________________
______________" }}{PARA 4 "" 0 "" {TEXT -1 26 "A. Propositions and Tab
les" }}{PARA 0 "" 0 "" {TEXT -1 83 "__________________________________
_________________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 187 "These will be the logical prop
ositions. They are essential binary valued functions which will help g
enerate the full range of possible coombinations of true and false in \+
our truth tables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 79 "Here is table of all of the possible permutgations of val
ues for p, g, r, and s" }}{PARA 0 "" 0 "" {TEXT -1 54 "A := array( [[ \+
p(k), q(k), r(k), s(k)] $ k = 0..15] );" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p := x -> irem(x, 2):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "q := x -> irem( iquo(x, 2
),2 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r := x -> irem( i
quo(x, 4),2 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "s := x ->
 irem( iquo(x, 8),2 ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 335 "We are using number for logical value - 0 for false
 and 1 for true. In this way we create a binary table which is equival
ent to a truth table. The first column has the values of p, the second
 column has the values of q, and so forth. Note that each row has a un
ique pattern of 0's and 1's that is not exactly the same as any other \+
row." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A := array( [[ p(k),
 q(k), r(k), s(k)] $ k = 0..15]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"AG-%'matrixG6#727&\"\"!F*F*F*7&\"\"\"F*F*F*7&F*F,F*F*7&F,F,F*F*7&F*
F*F,F*7&F,F*F,F*7&F*F,F,F*7&F,F,F,F*7&F*F*F*F,7&F,F*F*F,7&F*F,F*F,7&F,
F,F*F,7&F*F*F,F,7&F,F*F,F,7&F*F,F,F,7&F,F,F,F," }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 245 "With numbers we can see \+
the patterns easier and its also easier to use numeric functions to co
mpute the result. If you would like to see a customary truth table you
 can copy and paste the following instruction after each binary table \+
is created." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 229 "Truth_Table := A:\nfor i from 1 to linalg[rowdim](
A) do\nfor j from 1 to linalg[coldim](A) do\nif(  A[i,j] =1) then   Tr
uth_Table[i,j]  := 'True';\n                else Truth_Table[i,j] := '
False';\nfi; od; od; evalm(Truth_Table);    " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'matrixG6#727&%&FalseGF(F(F(7&%%TrueGF(F(F(7&F(F*F(F(
7&F*F*F(F(7&F(F(F*F(7&F*F(F*F(7&F(F*F*F(7&F*F*F*F(7&F(F(F(F*7&F*F(F(F*
7&F(F*F(F*7&F*F*F(F*7&F(F(F*F*7&F*F(F*F*7&F(F*F*F*7&F*F*F*F*" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 365 "This wil
l create a table with \"True\" and \"False\" in place of 0's and 1's. \+
If you prefer to see \"T\" and \"F\", you can change True to T and Fal
se to F in the commands above. For the remainder of this project we wi
ll create binary tables. Again, you can copy and paste the commands ab
ove after the creation of any binary table below to convert it to a tr
ue Truth Table." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 83 "_______________________________________________________________
____________________" }}{PARA 4 "" 0 "" {TEXT -1 26 "B. Negations and \+
Connetors" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________________
_______________________________________________________" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "We will also define thes
e special functions to perform logical operations. Make sure to capita
lize these names." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "NOT := x -> irem ( x + 1, 2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "AND := (x,y) -> irem ( x*y, 2):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "OR  := (x,y) -> max( x, y):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "XOR := (x,y) -> irem( x
+ y,2):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
159 "The negation operator, NOT, converts a logical expression to its \+
opposite : true to false, and false to true. Here is a table showing p
, q, not(p), and not(q)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 96 "Note that the first and third columns are opposite
 and so are the second and the fourth columns." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A := array( \+
[[p(k), q(k), NOT(p(k)), NOT(q(k)) ] $ k = 0..3] );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7&\"\"!F*\"\"\"F+7&F+F*F*F+7&F*F+
F+F*7&F+F+F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 82 "The connectives AND, OR, and XOR combine two logical prop
ositions in various ways." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 117 "Note that we are converting these logical stateme
nts into a functioin like notatioin to evaluate in the truth tables." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 3 "AND" }
{TEXT -1 75 " :                p and q = AND(p, q) is true only if bot
h p and q are true" }}{PARA 0 "" 0 "" {TEXT 258 6 "OR    " }{TEXT -1 
76 ":                p or q = OR(p,q) is true if either p or q or both
 are true." }}{PARA 0 "" 0 "" {TEXT 257 12 "Exclusive Or" }{TEXT -1 
69 ":    p xor q = XOR(p, q) is true only if p or q but not both are t
rue" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 68 "Lets see what the truth table looks like
 for these three connectives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 149 "The first column has the values of p. The seco
nd column has the values of q. The next columns have the values of (p \+
and q), (p or q) , and (p xor q)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "A := array( [[p(k), q(k), AND(p(k),q(k)), OR(p(k)),XO
R(p(k), q(k)) ] $ k = 0..3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
AG-%'matrixG6#7&7'\"\"!F*F*F*F*7'\"\"\"F*F*F,F,7'F*F,F*F*F,7'F,F,F,F,F
*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 4 "" 0 "" {TEXT -1 50 "C. Using Truth Table To Test Logi
cal Equivalencies" }}{PARA 0 "" 0 "" {TEXT -1 83 "____________________
_______________________________________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "One of the main
 values of truth tables is to test if two logical statements are equiv
alent" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 
"Lets test the proposition : not( p or q) = not(p) and not(q). We conv
ert the left and right sides of this equation into the function notati
on we defined above" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 148 "Since the two columns are exactly the same, the propos
ition is true. This is the abbreviated from of the truth table just sh
owing the final results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
82 "A := array( [[  NOT( OR(p(k), q(k) ),AND( NOT(p(k)), NOT(q(k)) ) )
] $ k = 0..3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#
7&7#\"\"\"7#\"\"!F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 149 "You can also include the original propositions and \+
intermediate steps. This makes it easier to see everything which contr
ibutes to the final results." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 149 "The columns are p, q, (p or q), not( p or q), \+
not(p) , and not(p)and not(q). To test the equivalency note that the 4
th and 8th columns are identical." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
130 "A := array( [[p(k), q(k), OR(p(k), q(k)), NOT( OR(p(k), q(k) )), \+
NOT(p(k)), NOT(q(k)), AND( NOT(p(k)), NOT(q(k))) ] $ k = 0..3] );" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7)\"\"!F*F*\"\"\"F
+F+F+7)F+F*F+F*F*F+F*7)F*F+F+F*F+F*F*7)F+F+F+F*F*F*F*" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 83 "_________________________________________________________
__________________________" }}{PARA 4 "" 0 "" {TEXT -1 50 "D. Using Tr
uth Tables To Test Logical Implications" }}{PARA 0 "" 0 "" {TEXT -1 
83 "__________________________________________________________________
_________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "An implication is writte
n either in words : if p then q, or symbolically, p -> q. A biconditio
nal is written p if and only if q, or p <-> q. We can define an implic
ation and biconditional function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "IF_THEN := (x,y) -> OR (NOT(
x), y):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "IF_ONLYIF := (x,
y) -> NOT( XOR(x,y) ):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 47 "We can create a truth table for the implication" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
64 "A := array( [ [p(k), q(k), IF_THEN( p(k), q(k)) ] $ k = 0..3] );" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7%\"\"!F*\"\"\"7
%F+F*F*7%F*F+F+7%F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 122 "How does the converse (if q then p) compare to th
e original implication (if p then q)?  Are they the logically equivale
nt?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "Yo
u can see they are different!   This means they are not logically equi
valent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "A := array( [[ IF_THEN( p(k), q(k) ), IF_THEN( q(k), \+
p(k)) ] $ k = 0..3] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'mat
rixG6#7&7$\"\"\"F*7$\"\"!F*7$F*F,F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 113 "How does the contrapositive (if not(a)
 then not(p)) compare to the origival implication? Are they the equiva
lent?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "
These are logically the same." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "A := array( [[ IF_THEN( p(k)
, q(k) ), IF_THEN( q(k), NOT(p(k))) ] $ k = 0..3] );" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7&7$\"\"\"F*7$\"\"!F*F)7$F*F," }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________
______________________________________________________________________
____" }}{PARA 4 "" 0 "" {TEXT -1 37 "E. Maple's Built - In Logic Funct
ions" }}{PARA 0 "" 0 "" {TEXT -1 83 "_________________________________
__________________________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 105 "Maple also has logic features built in which allow us to test \+
the validity of logical equations directly." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Here we use logical functions,
 and define two logical statements A and B. evalb() is used to evaluat
e an expression as a Boolean." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A := not(p or q);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG45%\"pG%\"qG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "B := not(p) and not(q);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"BG45%\"pG%\"qG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "evalb(A = B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr
ueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "
The evalb() command is used to test if the two statements are equivale
nt logically. Here are two more examples from above." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := not( \+
p or q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG45%\"pG%\"qG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B := not( p) and not( q);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG45%\"pG%\"qG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "evalb(A=B);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "A := p or not(q);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"AG5%\"pG4%\"qG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 
"B := not(p) and q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG34%\"pG%
\"qG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalb(A=B);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "A := ( a and b) or ( \+
c and d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG53%\"aG%\"bG3%\"cG%
\"dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "B := ( a or b) and \+
( c or d);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG35%\"aG%\"bG5%\"cG
%\"dG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalb(A=B);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 83 "________________________________________
___________________________________________" }}{PARA 4 "" 0 "" {TEXT 
-1 29 "F. Built - In Logic Functions" }}{PARA 0 "" 0 "" {TEXT -1 83 "_
______________________________________________________________________
____________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 555 "Above, when we considered logi
cal expressions, we are not assuming that the propositions p and q hav
e any particular values. In fact we want to make sure that two stateme
nts using p and q are the same for all combinations of values of p and
 q. In other cases, we might have propositions which are assigned valu
es and want to be able to work with them. You could make an analogy to
 algebra where you might work with variables wuch as x and y without h
aving any particular value, while at other times you might need to plu
g in particular values for x and y," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "We can d
efine variables to have logical values such as true or false." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "a := true;    b:= false;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG
%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG%&falseG" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "This can also be \+
done in an implied way." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 99 "In this case, 5 is not greater than 9, so a is false
, but 5 is indeed greater than 1, so b is true." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "a := (5 > 9)
;   b:= (5 > 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG2\"\"*\"\"&
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG2\"\"\"\"\"&" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "We can evaluate thre
e boolean variables using the " }{TEXT 259 5 "evalb" }{TEXT -1 9 " com
mand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalb(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalb(b);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 104 "We can also evaluate boolean expressions
. We can use the same operators and connectives ( not, or, and)." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }
{TEXT 260 5 "evalb" }{TEXT -1 107 " command evaluates the expression a
s a boolean (logical) expression and 'computes' a value of true or fal
se" }}{PARA 0 "" 0 "" {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "evalb( not(a));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%
trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf( a or b);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "evalf(a and b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&
falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{MARK "4 0" 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }