Boolean algebra may mean:
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Boolean algebra may mean:
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Boolean algebra is a specialized algebraic system which deals with boolean values, i.e. values that are either true or false. It forms part of a system called w:Boolean_logic, but we will discuss it here as part of a course on digital electronics.
Boolean algebra describes logical and set operations. A logical
operation might be for example: "I have flour and water, I can make
dough". In a case where I have flour and water, this statement is
true. If one of the elements is not true then it is clearly
false.
Another might be: "I have eggs or bacon, I have food" In a case
where I had eggs but not bacon, or if I had only bacon, or if I had
eggs and bacon, the statement I have food would be true. Only if I
did not have eggs nor bacon would "I have food" be false.
Boolean Algebra works like this. One creates statements which are true only if all their component statements are true.
Now, taking the first example, lets replace flour with a letter
representative of it: F, water with W, and dough with D. In order
to write it out we need a symbol for "and". The symbol for "and" in
boolean algebra is .
Therefore, bringing the above together, "I have flour and water, I
can make dough" could be described:
We now have the concept of symbols that can be true and false, and symbols describing logic. Lets have a look at these logic operations:
To add a bit of confusion, engineers often use + for OR and a multiply sign, x or * for AND. Some use a line above a symbol or expression to signify NOT, whereas others use the symbol ! after a term, or the symbol ~ before a term, to signify NOT. Examples:
Operations, One by one:
A truth table is a mathematical table which describes the output of a logical function in terms of its inputs for all combinations of different inputs.
AND

OR

You should get familiar with these because you'll be seeing a lot of them later.
We now know the symbols usable in boolean algebra, and what they mean, and have seen a few basic examples. Now lets have a look at rules and syntax:
So two elements contributing to one result is all very well, but
what if I consider bacon and eggs and Cheese food?
Elements can simply be chained together by putting in more
operations. So the above would be:
But what about if you have both OR and AND operations present? In
order to clearly and unambiguously communicate our equation, we
have to use parentheses(brackets). Bracketed expressions are solved
from the inner most brackets first. For example:
If I only consider only Bacon AND Eggs a meal, but cheese is also a
meal on its own I would write:
If I only consider Bacon AND Eggs, or Bacon AND Cheese a meal but
any one item alone, or egg and cheese not to be a proper meal I
would write:
Just as in ordinary algebra, where multiplication takes priority over addition, AND takes priority (or precedence) over OR. In fact, AND can be written as × and OR as +, and the instinctive notion of order of operations applies directly to Boolean Algebra.
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