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should do  | reasoning/proving/logic |  for the given code/algorithm/scheme

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either false (f) or true (t)
but not both

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"November 2, 2005 is a Thursday."
"At line 23 in this program, x is -2."
"My program runs faster than yours."
"Greased Lightning will win the race tomorrow." ?
"This sentence is false." ??
"What time is it?" ??

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   | 0 |  for false,  | 1 |  for true

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also incidentally, the restriction "not both" =  | law of excluded middle | 

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we build  | compound propositions |  a.k.a. propositional formulas

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 | logic |  operators a.k.a.  | connectives |  which are as follows:

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Engl.:   | not / negation | 

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Java:    | ! | 

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applies to one operand formula, which I call  | negand | , as follows:

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e.g.:  ! (s1.equals(s2))    // Java:   | ! (s1.equals(s2)) | 

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e.g.:  ! done               // Java:   | !done | 

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e.g.:  ! (x < y)            // Java:   | !(x < y) | 

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Engl.:   | and / conjunction | 

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Java:    | && / & | 

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applies to two operand formulas, called  | conjuncts | , as follows:

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e.g.:  (x < y) * (y < z)                // Java:   | (x < y) && (y < z) | 

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e.g.:  !done * (i < n)                  // Java:   | !done && (i < n) | 

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e.g.:  !(!done * i < n) * (c == x)      // Java:   | !(!done && i < n) && (c == x) | 

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Engl.:   | or / disjunction | 

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Java:    | || / | | 

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applies to two operand formulas, called  | disjuncts | , as follows:

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e.g.:  x<y / x&ge;y                                     &/&/ Java:   | x<y || x>=y | 

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e.g.:  (i<n * a[i]==null) / (!done * a[i]#null)      &/&/ Java:   | (i<n && a[i]==null) || (!done && a[i]!=null) | 

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e.g.:  !(i<n / a[i]==null) * (!done / a[i]#null)     &/&/ Java:   | !(i<n || a[i]==null) && (!done || a[i]!=null) | 

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Engl.:   | implies / implication / conditional | 

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Java:    | see an exercise (;-) | 

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    called  | hypothesis/premise/antecedent |  and  | conclusion/consequent | 

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e.g.:  x<y ? x+1&le;y                            &/&/ Java:   | see an exercise (;-) | 

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e.g.:  (i==n / done) ? ((i==n * done) / (!done * i<n))   &/&/ Java:   | see an exercise (;-) | 

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e.g.:  ((i==n ? done) * (done ? i==n)) / !done         &/&/ Java:   | see an exercise (;-) | 

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Engl.:   | biimplication / biconditional / equivalent | 

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Java:    |  == | 

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e.g.:  (x<y) &harr; (y>x)                   &/&/ Java:   | (x<y) == (y>x) | 

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e.g.:  !(x<y * y<z) = (!(x<y) / !(y<z))         &/&/ Java:   | !((x<y) && (y<z)) == (!(x<y) || !(y<z)) | 

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e.g.:  (!(x<y) = (x&ge;z)) ? (!((x<z) = (y&=z)) * x>y)    &/&/ Java:   | see an exercise (;-) | 

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Engl.:   | exclusive or / exclusive disjunction | 

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Java:    | ^ | 

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C++:    | ^ / xor | 

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one says it is a  | well-formed formula (WFF) | 

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interpret using  | precedences |  of operators e.g. "|&*|" evaluated before "|+|"

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when learned Java, learned  | semantics |  i.e. interpretations&/meanings  

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via  | truth table | 

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e.g. if |C1| stands for " | November 2, 2005 |  is a  | Thursday | ."

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  and if |C2| stands for "This room is  | MAK 1107 | ."

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is  |C1 &and; C2|  true&?  | no, it's false | 

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" | November 2, 2005 |  is a  | Thursday | , and this room is  | MAK 1107 | ."

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note |C1*C2 :=  | f | | generally,

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except |C1*C2 :=  | t | | when |C1 := t| and |C2 := t|

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note |D1/D2 :=  | t | | generally,

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except |D1/D2 :=  | f | | when |D1 := f| and |D2 := f|

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this is the case where |A| :=  | t | , |C| :=  | t | , |A?C| :=  | t | 

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this is the case where |A| :=  | t | , |C| :=  | f | , |A?C| :=  | f | 

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 | yes | 

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this is the case where |A| :=  | f | , |C| :=  | f | , |A?C| :=  | t | 

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 | yes, |z| could be |5| already | 

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this is the case where |A| :=  | f | , |C| :=  | t | , |A?C| :=  | t | 

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note |A?C :=  | t | | generally,

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except |A?C :=  | f | | when |A := t| and |C := f|

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e.g. |(i<n):= | t | , (y&ge;z):= | f | , done:= | t | ,| ...

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with |P:= | f | , Q:= | t | , R:= | f | |

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(! | f | ) / ( | t |  *  | f | )

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( | t  | ) / ( |    f    | )

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 |       t           | 

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values are specified for atoms: |P:= | f | , R:= | f | , Q:= | t | , S:= | t | |

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( | t |  ? ( | f |  * ! | f | ))  *  (( | f |  ? ( | f |  /  | t | )) *  | t | )

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( | t |  ? ( | f |  *  | t  | ))  *  (( | f |  ? ( |    t   | )) *  | t | )

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( | t |  ? ( |    f     | ))  *  (( |     t         | ) *  | t | )

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( |     f            | )  *  ( |                 t      | )

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 |                      f                           | 

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 | t    t   f     | 

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 | (  f (  f (t ))) f ((  t (  t  )) t  ) | 

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with |P:= | f | , Q:= | t | , R:= | f | | :

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P | Q | R || !P | Q * R | !P / (Q * R)
--+---+---++----+--------+-------------
f | t | f || t  |   f    |    t

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with |P:= | t | , Q:= | f | | :

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P | Q || ~Q | P -> !Q | (P -> !Q) -> Q
--+---++----+---------+---------------
t | f || t  |   t     |           f

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P | Q || ~Q | P -> ~Q | (P -> ~Q) -> Q
--+---++----+---------+---------------
t | t || f  |   f     |           t
t | f || t  |   t     |           f
f | t || f  |   t     |           t
f | f || t  |   t     |           f

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P | Q || P /\ Q | ~P | (P /\ Q) /\ ~P
--+---++--------+----+---------------
t | t ||   t    | f  |          f
t | f ||   f    | f  |          f
f | t ||   f    | t  |          f
f | f ||   f    | t  |          f

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P | Q | R || ~P | Q /\ R | ~P \/ (Q /\ R)
--+---+---++----+--------+---------------
t | t | t || f  |   t    |    t
t | t | f || f  |   f    |    f
t | f | t || f  |   f    |    f
t | f | f || f  |   f    |    f
f | t | t || t  |   t    |    t
f | t | f || t  |   f    |    t
f | f | t || t  |   f    |    t
f | f | f || t  |   f    |    t

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<li> a propositional formula is called a  | tautology |  or  | valid | 

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<li> a propositional formula that is always false is called a  | contradiction | 

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<li> two propositional formulas are  | logically equivalent | 

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P | Q || ~P | (~P) \/ Q
--+---++----+----------
t | t || f  |      t
t | f || f  |      f
f | t || t  |      t
f | f || t  |      t

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its  | converse |  is   |&kappa; &rarr; &alpha;| ,

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its  | inverse |  is   |&not;&alpha; &rarr; &not;&kappa;| ,

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and its  | contrapositive |  is   |&not;&kappa; &rarr; &not;&alpha;|

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atomic propositions, truth-values
logical connectives  /\  \/  ~  -->  (...)
  their truth tables
evaluation of formulas
truth tables of general formulas
terminology: equivalence, converse, contrapositive

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====================================================================

<p>
<hr>

<font size=-1>
(Copyright &copy; 2009 by
Hugh McGuire
&nbsp;
&mdash;
<cite>for thoughts about this, see </cite>
&nbsp;
<a href="/~mcguire/teaching/copyright_thoughts.html">
<kbd>http://www.cis.gvsu.edu/~mcguire/teaching/copyright_thoughts.html</kbd></a>&nbsp;.)
</font>