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The Computer Inside Out: Boolean Algebra
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Boolean Algebra
The binary system allows for a new system of mathematics. In
the mid-19th century, George Boole devised such a mathematical
system (long before computers were created). The mathematics that
he made up are now called boolean algebra. Boolean algebra is
used all the time in programming, and is thus very useful to
know. At the bottom of the page, we have prepared an applet to
show results of boolean operations.
Boolean algebra contains 4 operations: AND, OR, XOR, and
NOT.
- AND
Results in a 1 if both of the bits are 1. Else results in a
0.
A AND B is commonly expressed as AB or A*B.
Truth table:
| A |
B |
X (A AND B) |
 |
| 0 |
0 |
0 |
| 0 |
1 |
0 |
| 1 |
0 |
0 |
| 1 |
1 |
1 |
- OR
Results in a 1 if either or both of the bits are 1. Else
results in a 0 (if both bits are 0).
A OR B is commonly expressed as A+B.
Truth table:
| A |
B |
X (A OR B) |
 |
| 0 |
0 |
0 |
| 0 |
1 |
1 |
| 1 |
0 |
1 |
| 1 |
1 |
1 |
- XOR (eXclusive OR)
Results in a 1 if the two bits are different, and a 0 if they
are the same.
A XOR B is commonly expressed the same way as A OR B, except
that the plus sign has a circle around it.
Truth table:
| A |
B |
X (A XOR B) |
 |
| 0 |
0 |
0 |
| 0 |
1 |
1 |
| 1 |
0 |
1 |
| 1 |
1 |
0 |
- NOT
This is a unary operation (like a negative sign). NOT results
in the opposite of the bit.
NOT A is commonly expressed as A with a bar on top.
Truth table:
| A |
X (NOT A) |
 |
| 0 |
1 |
| 1 |
0 |
Although we have not presented them, there are other
operations such as NOR, XNOR, and NAND. These particular ones are
simply the NOT alternative of OR, XOR, and AND, respectively. The
electronic sign for these is the same as what was presented
above, except that there is a small circle on the tip, such as
the one shown in the NOT picture.
To give you some experience about how all of these operations
work, we have written a java applet which is able to execute all
of the above boolean operations. However, the applet will be able
to do boolean operations on multi-bit numbers. The only
difference between that and what we showed you above is that
above we used single digits, while with multiple digits the
operation is performed on each matching pair of digits (i.e. the
first bit of A with the first bit of B, and etc). Since NOT only
takes one input, the first input is used when the NOT function is
performed.
Note: You will need a Java 1.1 compliant
browser, such as IE 4.0+, and Netscape 4.5+.
Now that you understand everything that there is to know about
Boolean algebra, you can go on to the mathematics section.
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