George Boole is famous because he showed that rules used in the algebra
of numbers could also be applied to logic. This logic algebra, called
Boolean algebra in his honor, has many properties which are similar to
"regular" algebra. These rules can be very helpful when
we are asking the question such as "Is the expression **ABC+D**
the same as **(A+D)(B+D)(C+D)**?" They can also help us to reduce
an expression to an equivalent expression that has fewer operators.

These properties, called "laws", come in two flavors: the
*conjunctive form* and the *disjunctive form*. This section discusses
each of these laws and, in many cases, proves them when they are not obvious.

**The Idempotent Laws:**These laws express the fact that a Boolean value, when*or*ed or*and*ed with itself, remains the same.

**The Associative Laws:**These laws show that when you have more than two operations of the same kind in a row, it doesn't matter the order in which they are performed.

**The Commutative Laws:**These laws show that the order in which you write down the operands of a Boolean operation is not important.

**The Distributive Laws:**These laws show that combinations of*and*and*or*can be written in an expanded form.

**The Identity Laws:**These laws show that a single variable interacts with the constant values TRUE (shown here as*T*) and FALSE (shown here as*F*) with consistent results.

**The Complement Laws:**These laws define how value combines with its complement.

**The Involution Law:**There is only one form of this law, which addresses double complementation.

**DeMorgan's Law:**We proved this law in section 2. It shows the relationship between*and*and*or*when complementation is applied.

We will demonstrate how these laws can be used to convert one expression
into another, equivalent expression. Consider a new operation known as
*exclusive or*, also called *xor*. Exclusive or is TRUE if
and only if exactly one of the operands is TRUE. The truth table
for exclusive or is:

A |
B |
A xor B |

0 |
0 |
0 |

0 |
1 |
1 |

1 |
0 |
1 |

1 |
1 |
0 |

Can we write a Boolean expression using only disjunction, conjunction and complementation which computes exclusive or? Yes. The expression follows directly from the middle two rows of the truth table:

This expression requires 5 operations to compute: two complements, two disjunctions, and one conjunction. Can we find an equivalent expression that uses fewer operations? Using our laws of Boolean algebra, and a little cleverness, the answer is "yes." Each of the following shows a step in the transformation of the above expression to a simpler one.

- This step is possible because
of the identity law.
*or*ing more FALSE terms with the ones we already have will not change the overall value.

- This step is possible because
of the complement law. The FALSE terms can be turned into conjunctions
of a variable with its own complement.

- This step is possible with
a reverse application of the distributive law.

- This is the final step, which
was achieved by application of DeMorgan's law. This expression only has
four operations, yet is equivalent to the original. It also "makes
sense" when read: "Exclusive or is true when either
**A**or**B**is true, but not both!"

- Prove each of the laws of Boolean algebra by giving appropriate truth tables. You do not need to prove DeMorgan's law, since that was done in section 2. Use that as an example of how to prove the others.
- Can you reduce the following expressions?

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Logic Gates
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