Lecture 9: Branching II / Boolean Algebra

George Boole (1815-1864) English mathematician. His work "The mathematical analysis of logic"  (1847) established the basis of modern mathematical logic, and his boolean algebra can be used in designing computers. Boole's system is essentially two-valued. This can be symbolized by 0     or   1   "binary representation" TRUE  or FALSE  "truth representation" 0 V   or  5 V   "TTL electronics (transistor-transistor logic)" 0 pC  or  1 pC  (pC = pico-Coulomb)  "the charge in a condensator, the elementary memory unit in (dynamic) RAM"

In the previous lecture (aula 8) we have seen how we can control the flow of the program by the branching instructions if ... then and if ... then ... else. We used conditions like (x<1.0). Now imagine we apply this to calculate the square-root of (x²- 4). Clearly, this doesn't have an answer for x between -2 and +2. It would be nice to check if x is in this range or not. We could solve this with
  if (x<2) then
    if (x>-2) then
      WriteLn('Error');
Much nicer would be if we could do this in a single condition. We will now see that exactly that is possible
  if ((x<2) AND (x>-2)) then WriteLn('Error');
which obviously mean that both conditions (x<2 and x>-2), should be true for the complete condition to be true.
This is an example of a Boolean calculation
  condition3 := condition1 AND condition2;
  if condition3 then instruction;
Other Boolean algebra operators that can be used in PASCAL are OR and XOR. (Others that are not implemented include NAND and NOR). The AND and OR operations are obvious. The XOR operation is a little complicated. a XOR b means "a or b true, but not both!"
Finally, there is the boolean negator NOT. It means taking the opposite; if a is TRUE, NOT a is FALSE, and vice cerse. Wheras AND, OR and XOR need two operands (for example a XOR b), NOT needs only one (NOT a).
With these four operators we can calculate all possible conditions we will need.

For completeness sake, here is the complete calculation tables ("truth tables") for the four Boolean operators used in modern programming languages.
 

Boolean operator  AND OR XOR NOT	AND a b  a AND b  TRUE TRUE TRUE TRUE  FALSE  FALSE  FALSE  TRUE FALSE  FALSE  FALSE FALSE	OR a b a OR b TRUE TRUE TRUE TRUE FALSE TRUE FALSE TRUE TRUE  FALSE   FALSE   FALSE	XOR a b  a XOR b  TRUE TRUE FALSE TRUE FALSE TRUE FALSE TRUE TRUE  FALSE   FALSE   FALSE
	NOT a NOT a TRUE  FALSE   FALSE  TRUE

Examples:

a := 1;
b := -2;
c := 3;
d := 3;

if (a>0) then
  WriteLn('TRUE')
else
  WriteLn('FALSE');

(a>0)                     TRUE
(b>0)                     FALSE
(a>0) AND (b>0)           FALSE
(a>0) OR (b>0)            TRUE
(a>0) XOR (b>0)           TRUE
NOT (a>0)                 FALSE
(NOT (a>0)) OR (b>0)      FALSE
(2=b) OR (NOT (2=b))      TRUE
(NOT (c>0)) XOR (d>0)     TRUE

  PROGRAM Years;

  VAR ano: integer;

  begin
    WriteLn('From what year are you?');
    ReadLn(ano);
    if (ano=1) then
      writeln('primeiro ano: MAT-1, CALC-1')
    else
      if (ano=2) then
        writeln('segundo ano: INF, LIN-ALG')
      else
        if (ano=3) then
          writeln('terceiro ano: ELEC, FYS')
        else
          if (ano=4) then
            writeln('quarto ano: QUI, MAT-2')
          else
            if (ano=5) then
              writeln('quinto ano: PROJECTO')
            else
              writeln('>5: AINDA NAO ACABOU?');
  end.

(Note the structure of the program, with indentations. Also note the missing ; before every else).
This program will run without problem
From what year are you?
 1
primeiro ano: MAT-1, CALC-1

From what year are you?
 4
quarto ano: QUI, MAT-2

The structure is not very readable, though. To make it better, we can use case ... of. Whereas in "if condition then instruction1" the condition is necessarily of the boolean type (TRUE or FALSE), in case ... of we can use any type of variable that has discreet values (in contrast to floating point variables which do not have discreet, whole number values).
 

The expression needs to result in a value of any countable type (for example: integer, byte, boolean, but also char. Not, for example: real or string). This can be a simple variable or a calculation resulting in a value. The values value1 to valueN also have to be of the same type;
As an example the above program rewritten to make use of case ... of:

  PROGRAM Years;

  VAR ano: integer;

  begin
    WriteLn('From what year are you?');
    ReadLn(ano);
    Case ano of
      1: writeln('primeiro ano: MAT-1, CALC-1');
      2: writeln('segundo ano: INF, LIN-ALG');
      3: writeln('terceiro ano: ELEC, FYS');
      4: writeln('quarto ano: QUI, MAT-2');
      5: writeln('quinto ano: PROJECTO')
      else writeln('>5: AINDA NAO ACABOU?');
    end;
  end.

This progam will have the same output as the program before, but now it is more readable.

Aula 9:    (2*B) OR (NOT (2*B))