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 <FONT SIZE=5><B> Boolean logic</B></FONT><BR>
 <FONT SIZE=4>Operations AND, OR, XOR and NOT </FONT>
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<P>
A bit is the minimum amount of information that we can imagine, since it only stores
either value 1 or 0, that represents YES or NOT, activated or deactivated, true or false,
etc... that is: two possible states each one opposite to the other, without possibility
of shades.
<P>
Several operations can be performed with bits, either in conjunction with other bits or
themselves alone.  These operations receive the name of <B>boolean operations</B>,
a word that come from the name of one of the mathematicians who more contributed to
this field: George Boole (1815-1864).
<P>
All these operations have a predetermined behavior and all of them can be applied to any
bit being anyone the value contained in it (<TT><B>0</B></TT> or <TT><B>1</B></TT>).
Next you have a list of the basic boolean operations and a table with the behaviour
of that operation with every possible combination of bits.
<P>
<DL>

<DT><B>AND</B><BR>
<DD>this operation is performed between two different bits
(<tt><B>a</B></TT> and <TT><B>b</B></TT>). The result is <TT><B>1</B></TT> if both
<TT><B>a</B></TT> and <TT><B>b</B></TT> are equal to <TT><B>1</B></TT>, if anyone of them
is equal to <TT><B>0</B></TT> the result is <TT><B>0</B></TT>
<blockquote ><B>AND (<TT>&amp;</TT>)</B>
<TABLE BORDER=1 CELLSPACING=0>
<TR><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><TT><B>a</B></TT></TD><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><B><TT>b</TT></B></TD><TD BGCOLOR="#8030FF" WIDTH=50 ALIGN="center"><B><TT>a&amp;b</TT></B></TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
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</BLOCKQUOTE>

<DT><B>OR</B><BR>
<DD>this operation is performed between two different bits
(<tt><B>a</B></TT> and <TT><B>b</B></TT>). The result is <TT><B>1</B></TT>
if any of them <TT><B>a</B></TT> or <TT><B>b</B></TT> is <TT><B>1</B></TT>.
If noone is equal to <TT><B>1</B></TT> the result is <TT><B>0</B></TT>
<blockquote ><B>OR (<TT>|</TT>)</B>
<TABLE BORDER=1 CELLSPACING=0>
<TR><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><TT><B>a</B></TT></TD><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><B><TT>b</TT></B></TD><TD BGCOLOR="#8030FF" WIDTH=50 ALIGN="center"><B><TT>a|b</TT></B></TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
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</BLOCKQUOTE>

<DT><B>XOR</B> (Exclusive Or)<BR>
<DD>this operation is performed between two different bits
(<tt><B>a</B></TT> and <TT><B>b</B></TT>). The result is <TT><B>1</B></TT>
if any of them <TT><B>a</B></TT> or <TT><B>b</B></TT> is <TT><B>1</B></TT>
except if both they are it. If neither or both are equal to <TT><B>1</B></TT> the result
is <TT><B>0</B></TT>
<blockquote ><B>XOR (<TT>^</TT>)</B>
<TABLE BORDER=1 CELLSPACING=0>
<TR><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><TT><B>a</B></TT></TD><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><B><TT>b</TT></B></TD><TD BGCOLOR="#8030FF" WIDTH=50 ALIGN="center"><B><TT>a^b</TT></B></TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
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</BLOCKQUOTE>

<DT><B>NOT</B><BR>
<DD>This operation is performed on a single bit.
Its result is the inversion of the present value of the bit:  if it
contained <TT><B>1</B></TT> passes to contain
<TT><B>0</B></TT> and if it contained <TT><B>0</B></TT> it passes to contain
<TT><B>1</B></TT>:
<blockquote ><B>NOT (<TT>~</TT>)</B>
<TABLE BORDER=1 CELLSPACING=0>
<TR><TD BGCOLOR="#8080FF" WIDTH=50 ALIGN="center"><TT><B>a</B></TT></TD><TD BGCOLOR="#8030FF" WIDTH=50 ALIGN="center"><B><TT>~a</TT></B></TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">0</TD><TD BGCOLOR="silver" ALIGN="center">1</TD></TR>
<TR><TD BGCOLOR="white" ALIGN="center">1</TD><TD BGCOLOR="silver" ALIGN="center">0</TD></TR>
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</BLOCKQUOTE>

</DL>

<P>
These are the 4 basic boolean operations (AND, OR, XOR and NOT). Combining these operations
any desired result can be obtained.
<P>
In C++, these operators can be used between two variables of any integer data type; 
the logic operation is made between the bits of the two variables.
For example, suposing two variables: <TT><B>a</B></TT> and <TT><B>b</B></TT>,
both of type <TT><B>char</B></TT>, <TT><B>a</B></TT> contains <TT><B>195</B></TT>
(<TT><B>11000011</B></TT> in binary) and <TT><B>b</B></TT> contains <TT><B>87</B></TT>
(or <TT><B>01010111</B></TT> in binary).  If we write the following code:
<blockquote ><TT>
char a=195;<BR>
char b=87;<BR>
char c;<BR>
c=a&b;
</TT></BLOCKQUOTE>
That means, that we conducted a bitwise <TT><B>AND</B></TT> operation between
<TT><B>a</B></TT> and <TT><B>b</B></TT>. The result (the content of <TT><B>c</B></TT>)
would be the following one:
<blockquote ><IMG SRC="boolean1.gif"></BLOCKQUOTE>
<TT><B>01000011</B></TT>, that is <TT><B>67</B></TT>.

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